The Kido Butai (1st Air Fleet) consisted of the 1st, 2nd and 5th Carrier Divisions at the onset of the Pacific war. 2nd Carrier Division was originally slated by the Japanese Navy General Staff to cover the Southern Operations (Malaya and NEI) but that was changed when Yamamato threatened to resign if he didn't get 6 carriers for the Pearl Harbor operation. I can't find anything that states specific missions for the rest of the Kido Butai prior to their being tasked to the Pearl Harbor ops but it appears they were to be tasked to support the southern operation. The only decent reference is in Bartsch, "December 8, 1941, MacArthur's Pearl Harbor", page 96: "However, the navy General Staff - responsible for planning naval operations- regarded Yamamato's Hawaii operation as dangerous and having little hope of success. It wanted to reserve all of the Combined Fleet's carriers for southern operations, not to split them between the two."
The remaining carriers of the IJN, the 3rd Carrier Division (Zuiho) and 4th Carrier Division (Ryujo and Taiyo) were assigned to cover the Philippine Island operations.
If the Pearl Harbor operation does not happen, parsing out a division of the Kido Butai to cover the Wake Island landings goes in the face of the current IJN doctrine for the Kido Butai. It was envisaged as working together to overwhelm enemy airpower. They practiced coordinated attacks with one division providing fighter protection while another division would provide the bombing aircraft; their second wave would flip-flop the division's duties (Look into "Kaigun" by Evans and Peattie to get a more detailed explanation.) Splitting up this force just goes against everything they had been working at since its formation in April 1941. Secondly, the Wake Island landings were a sideshow not a major objective. It just doesn't make sense to allocate carriers to cover this operation. The IJN would have used them to either support the main operations to the south or kept them with the Combined Fleet ready to respond to any USN offensive actions.
Again, the concept of the Decisive Battle was for the outlying islands to attack with their aircraft to weaken the advancing USN forces, the same would be done with the submarine force. The area for the Decisive Battle had advanced over the years from off the coast of Japan to just west of the Marianas and Yamamato had even proposed conducting it in the waters east of the Marianas. But the concept remained the same, to wear down the USN forces with air and submarine attacks, night attacks by light forces using torpedoes and then finally the Decisive Battle.
This is probably one of, if not THE best, responses I have ever read, as it is loaded with facts and information, and even has mention of reference works that I can peruse at my leisure! I want to give a well earned shout out to
@Gunner's_Quadrant for such a well done and informative reply.
When I first looked at the OP for this thread, I started looking at the map and asking myself, what else would the Japanese attempt on Dec 7th, if Oahu was off the table, and then realized that either they would have to use the KB in the south, or else at Wake. Wake is not an important enough target to warrant the full attentions of the entire KB (as you yourself pointed out), and we know then that either the entire KB maintains cohesion as a single force, or is broken up and used in separate battles in multiple other places. Given that the Japanese were not ones to let forces sit around and do nothing, I'm thinking that their single most potent force is going to be USED right from day one, so that rules out sitting back and doing nothing rather convincingly, so it's either Wake, as a single force, or various operations in the SRA, which means breaking up their force.
When I thought about that, I then had to come up with a rational for either course of action, and the southern resource area has far to many potential uses for the KB to be able to lend a hand, and I concluded that if they were sent south, they would be broken up and used in as many places as possible, right off the bat and until they had won the theater entirely, which would entangle them for perhaps as much as six months, and potentially leave themselves open to USN counter attacks, and the countering of which might throw off their expansion operations when suddenly deprived of the customary (but unneeded, at least to my mind) carrier support.
I then looked at what could possibly be worth the attentions of the KB to the East of North, and came up with something that, while not offering the long shot hope of a quick and glorious attack on Oahu, would at least offer up the potential to give Japan a shot at a better initial position from which to go on the wait, attrition, and then crush the USN Pacific fleet.
For your consideration, let us say that Yamamoto decides to swarm Wake with the overwhelming power of the KB, invade and secure, and get Japanese land based air in-placed ASAP, and then hit Midway island with the full force of the KB as quickly as possible thereafter, on the premise that the US defenses at midway are only going to be getting stronger and stronger the longer they are left alone, and that taking out Midway's air power is a logical next step in extending (and filling the gaps in their defensive perimeter), while maintaining force concentration in case the USN makes a mistake and attempts a half-assed intervention with a force to weak to fight toe to toe with all 6 carriers of the KB.
Yes, this is a risk (although we can surely agree, much less of a risk than OTL PH), and gives the Japanese the options to invade Midway (and lets face it, the Western Aleutians, as well) far earlier than in OTL, and allows the Japanese to emplace land based air and seaplanes forward earlier.
The US is then looking at the front lines being both the Hawaiian & Aleutian islands, forcing them to divert aircraft to these theaters even if the Japanese never attempt to push further East, and this takes away aircraft that could otherwise have been sent to Australia/Philippines.
Anyway, that is what I was able to come up with in the short term, to Justify the KB NOT hitting PH, and yet still doing something.