The argument goes like this: The Quran tells us prophet Muhammad was given the book (al kitab) and the wisdom (al hikma) and they say “al kitab” is The Quran and “al hikma” is separated by “wa” (and) so must be something else, which they take to mean the prophet’s sunna. Thus using this as a Quranic justification for following the prophet’s sunna, which they claim is documented in traditions (e.g. hadith), thus making those traditions a valid source of law.

There are a number of assumptions in this argument:
1) “al kitab” means the book
2) the kitab/book refers to The Quran
3) the conjunction “wa” (and) means a separate and distinct thing
4) “al hikma” refers to prophet Muhammad’s sunna
5) this sunna is preserved/documented accurately
6) this documented sunna is authoritative i.e. must be obeyed

This article will address points 1-4.

Firstly, it should be noted according to traditional Muslim history The Quran was compiled into a book after prophet Muhammad’s death. So if “al kitab” means the book why does The Quran regularly refer to itself as "al kitab" when it wasn't a book at that time? Traditionalists explain this by saying even a chapter or a verse can be referred to as a book or it is an ongoing book or that it is preserved as a book in the guarded tablet mentioned in 85:22. Some may find these explanations satisfactory.

In modern Arabic “kitab” often means book however in Classical Arabic (the language of Al Quran) it is not necessarily the case. The verbal noun can mean "he wrote/prescribed/decreed/appointed/recorded/collected it". It does not necessarily mean written down (as in a physical copy), it could be but doesn’t have to be. Thus some sort of authoritative instruction may be a more apt general meaning e.g. ‘decree’. The definite article "al" before the word "kitab" usually denotes specificity, e.g. something more well known to the audience. ?

In these verses it is common for translators to use other than “book” (2:235, 3:145, 4:24, 4:103, 6:59, 8:68, 8:75, 9:36, 10:61, 13:38-39, 15:4, 17:13, 17:58, 21:104, 23:62, 24:33, 27:28-29, 30:56 and others) some examples shown below:

2:235 ...and do not confirm the knot of marriage until the prescribed term (kitab) reaches its term
3:145 and not is for a soul that it dies except by permission of God at a decree (kitab) determined
4:103 ...indeed the salat is on the believers a timed decree (kitab)
8:68 had not an ordainment (kitab) from God preceded surely would have touched you a great punishment...
9:36 number of months with God is twelve months in the kitab (of) God (from) day He created heavens and the earth...
13:38 ...for everything is a term prescribed (kitab)...
15:4 ...but there was for it a decree known...
17:7 ...a record (kitab) which he will find unrolled, read your record (kitab)...

21:104 the day We will fold the heaven (like) folding a scroll for records/kutub (plural of kitab)...*
*it may be interesting to consider that the universe can be thought of as God's canvas/scroll containing His writings/decrees/records. Note if it was translated as "scroll for books" it doesn't fit the common conception of what a book looks like, same for 17:7 above.
24:33 ...and those who seek the writing/decree (al kitab) from those who possess your right hands then give them the writing/decree (verb: katibu) if you know in them any good...
27:28 go with my letter/decree (kitab) this and deliver it to them...
98:2-4 – a messenger from God reciting pages purified wherein (are) upright writings/decrees/kutub (plural of kitab)

 If it is argued "kitab" inherently meant something written then it may be redundant to describe it as written/inscribed in verses 17:58, 33:6, 52:2.

As a side note "people of the book” is commonly taken to be the Jews/Christians but of course they do not follow a single book. According to Traditionalists the Jews have their own and as do the Christians, so some may say using the singular book/kitab doesn’t quite fit.

The above is not considered to be strong enough evidence to prove "kitab" does not mean "book" in The Quran but it is sufficient to show that it has varied usage. It could be argued within context of revelation it means book and we will check this later. Also I personally did not find one occurrence of its many occurrences wherein it definitely meant book, but if others are aware of such please let me know.

Variance among translators with regard to what is "al kitab" here when it suggests everything is recorded in "al kitab":
6:38 ...nothing We have neglected in the kitab from anything...
6:59 ...but is in a clear kitab
10:61 ...but is in a clear kitab
11:6 ...all in a clear kitab
Most traditional exegetes say this is referring to a master record with God (85:22) not The Quran. Side note: see similar use of "imaami" in 17:71 and 36:12.

The following clearly shows that in the context of revelation "al kitab" and "al quran" are different:
10:37 and not this the quran could be produced by other than God but is a confirmation of which is between your hands and a distinguishing/detailing* (of) the kitab no doubt in it...
*to make distinct/clear, to set apart.
The connotation of the word "tafsila" is to make distinct / delineate / set apart which could suggest The Quran is a part of "al kitab". This suggestion is confirmed by The Quran elsewhere. Note the use of the partitive "min" (of/from, i.e. a part of):

18:27 and recite what has been revealed to thee of the kitab of your Lord...
35:31 and that which We have revealed to thee of the kitab it is the truth confirming what was between your hands...
29:45 recite what has been revealed to you of the kitab...
2:174 indeed those who conceal what God has revealed of the kitab...
2:231 ...remember the favours of God upon you and what is revealed to you of the kitab and the wisdom/hikma, He instructs you with it...

Concept explicitly mentioned here:
3:23 have you seen those who were given a portion of the kitab? They are invited to kitab of God that it should judge between them...
Also see 4:44 and 4:51.

Also evidenced here:
43:1-4 Ha Meem, by/and the clear/manifest kitab, indeed We have made it a reading in Arabic so that you may understand, and indeed it is in the mother/source (of) the kitab with Us, surely exalted full of wisdom (see 13:39 for similar)
56:77-78 Indeed it is surely a quran/reading noble in a kitab well-protected

Taking "al kitab" as "al quran" here wouldn’t make much sense, so exegetes take 1st occurrence of "al kitab" as "al quran" and the 2nd occurrence "al kitab" (singular) to mean scriptures (plural) e.g. Tawrat/Injeel/etc, to make it fit.
5:48 and We have revealed to thee the kitab with truth confirming what is between your hands of the kitab and a guardian over it...”

After mentioning various prophets it says:
6:89 those are ones whom We gave them the kitab and the hukm/judgement and the prophethood so if disbelieve in it these then indeed We have entrusted it to a people who are not in it disbelievers.
6:90 Those are ones whom God guided so of/in their guidance you follow
Note: Despite mentioning many prophets, in Traditional Islam only 4 other books have been given to prophets (Abraham, David, Moses, Jesus), so I assume traditionalist exegetes will claim 6:89 only refers to those prophets.

Similarly:
3:81 And God took a covenant from the prophets: "For what I have given you of the kitab and hikma/wisdom, then a messenger will come to you authenticating what is with you. You will believe in him and support him...
Did they all receive one kitab/book? No-one thinks this to my knowledge. Did all prophets receive a book according to Traditional Islam? No-one thinks this to my knowledge. Tafsir ibn Kathir says "for what I have given you of the kitab and hikma" means "if I give you the book and the hikma" - how convenient!

The following shows "Wa" does not always mean a separate different/distinct thing:
55:68 in both of them are fruit AND date-palms AND pomegranates
Date-palms and pomegranates are also fruit.
33:7 ...We took from the prophets their covenant AND from you AND Noah and Abraham and Moses and Jesus...
The people mentioned are also prophets.
2:98 whoever is an enemy to God and His angels and His messengers AND Jibreel AND Meekael then...
Jibreel and Meekael are also considered angels.
15:87 ...We have given thee seven of the oft-repeated AND the great quran
Most exegetes consider "seven of the oft-repeated" to be chapter 1 (al-fatiha) but it is still considered part of Quran.
2:238 guard the salawat (plural) AND the wusta/balanced salat...
"wusta salat" is already included in salawat according to Traditionalists.

There are likely many more examples but the above should be sufficient to counter the Traditional argument.

If the argument is "wa" means a separate and distinct thing then do these examples weaken the "al kitab" is "al quran" argument?
15:1 A L R these are the signs/verses  (of) the kitab AND a clear/manifest reading/quran...
27:1 T S these are the signs/verses (of) the quran AND a clear/manifest kitab
If "wa" means a separate and distinct thing then use of "kitab" above cannot be referring to "quran" thus creating a conflict in their view. Choosing "kitab=quran" will invalidate their "wa" claim, and choosing "wa" separates distinct things will invalidate their "kitab=quran" claim.

This paired expression (al kitab wa al hikma) is mentioned 9 times in Quran (2:129, 151, 231, 3:164, 4:54, 113, 62:2, 3:48, 5:110). The last two references are related to Jesus and it is mentioned along with the Tawrat and Injil. This creates an obvious problem for Traditionalists because if they are taken as 4 separate sources (book/kitab + hikma + tawrat + injil) they do not teach Jesus received another scripture apart from the Injil. We can see from Traditional tafsirs how they try to get around this problem with their common tactic of being inconsistent, e.g.
Tafsir Al Jalalayn translates it as “He will teach him the Book that is script wisdom and the Torah and the Gospel”
Tafsir ibn Kathir “and when I taught you the Book and the Hikmah (the power of writing and understanding) and the Tawrah and the Injil...”
If they were to be consistent and interpret it the same way when it is used with prophet Muhammad it would obviously refute their position and/or cause other problems. Contradiction/inconsistency is a hallmark of man-made works, see the important test Quran 4:82.

Are all the following examples of separate revelation types? If so what are they and how are they different:
4:113 ...and God has sent down to thee the kitab AND the hikma AND taught you what you did not know
Can the above be 3 separate things? 42:52 makes it clear the messenger did not know the kitab before thus would fall under "what you did not know" proving beyond doubt the "wa" can be relational/linked to what came before or as some put it a clarifying/specifying "wa".

Are all these separate types of revelation?
2:185 ...ramadhan is that was revealed in it the quran a guidance for mankind AND clear proofs of the guidance AND the criterion/furqan (also see 3:3-4, 25:1)
3:58 that is what We recite to you of the verses/ayat AND the wise reminder
3:184 messengers from before you came with the clear signs AND the zubur/scriptures AND the enlightening kitab
Note "zubur" is plural. According to traditional Islam did David come with the Zabur AND the enlightening kitab/book (i.e. two separate books)? Not to my knowledge.
57:25 We sent our messengers with the clear proofs AND We sent down with them the kitab AND the mizan/balance that may establish the people with justice
42:17 God is the one who sent down the kitab with truth AND the balance/mizan...
21:48 We gave Musa and Aaron the furqan/criterion AND a light AND a reminder for the righteous (also see 2:53)
16:125 Call to the way of your Lord with the wisdom AND the good instruction...
add 38:20 ?

To my knowledge treating them (e.g. al furqan, al mizan, al dhikr etc) as separate types of revelation, as claimed for kitab and hikma, is unheard of in Traditional tafsirs. If the reader is aware of this in Traditional sources please let me know.

As can clearly be seen from the above the Traditional position is weak, and is inconsistent at best and contradictory at worst.

The above argument necessitates that the wisdom/hikma being spoken of is external to The Quran. However the revelation itself is described as wise and having wisdom:
17:39 that is from what was revealed to thee from your Lord of the wisdom...
36:2 ...the wise quran/reading
31:1-3 A L M these are the signs/verses of the wise kitab
3:58 that (is what) We recite to thee of the verses/signs and the wise reminder
43:1-4 Ha Meem, by/and the clear/manifest kitab, indeed We have made it a reading in Arabic so that you may understand, and indeed it is in the mother/source (of) the kitab with Us, surely exalted full of wisdom
44:1-4 H M and/by the clear/manifest kitab indeed We revealed it(m) in a night blessed indeed We are (ever) warning, in it(f) is divided every wise/hakim matter
54:4-5 and certainly has come to them of the information wherein is deterrence wisdom reached/perfected

Some of the above will depend on how one interprets it but at the very least we can say some wisdom/hikma can be found in The Quran and it is an attribute of Quran, so it's possible it could be referring to that.

In Quran "hikma" is not exclusive to prophet Muhammad, as many were given wisdom, e.g. 3:81 (prophets), 3:48 / 5:110 / 43:63 (Jesus), 19:12 (Yahya as a child), 26:83 (Abraham), 28:14 (Moses), 2:251 and 38:20 (David), 4:54 (family/people of Abraham), 45:16 (children of israel), 31:12 (Luqman, note that Traditionalists do not regard him as a prophet e.g. see tafsir ibn kathir).
Some Traditionalists may claim when hikma/wisdom is used with previous prophets it is referring to their “sunna” e.g. sunna of Abraham, sunna of Moses, sunna of Jesus etc however we have no evidence of this whatsoever. I have personally never read an article or heard a lecture from a Traditional scholar talking about other prophets and their sunna, not to mention the preservation of such sunna (since they would presumably consider it a critical source of guidance). One sign that often exposes a future made-up view is that there is no trace of discussion in the past.
The example of Abraham's people/family being given wisdom (4:54) is interesting because Abraham's people/family would have lived at the same time or their timelines overlapped thus according to Traditionalists did they individually receive a sunna, or does it mean they were following Abraham's sunna? Similar could be asked for children of israel (45:16). Does this make sense to you?

Quran implies anyone could be granted hikma/wisdom:
2:269 He grants the hikma/wisdom to whom He wills and whoever is granted the hikma/wisdom then certainly he is granted abundant good and none pays heed except those of understanding
Interestingly tafsir ibn kathir says "hikma" here is knowledge of Quran. There is variance in wording when it comes to the hadith they reference for this understanding, some hadith explicitly relate it to Quran and some seem to imply knowledge/wisdom in general.