For CAs, they will *always* be faster than an equally teched carrier.
But for fun let's look at the IC cost and sea attack we can get out of them.
Assuming 1940 techs for pearl harbor analogue.
If you spend the same K/icd as the BB + dd fleet (roughly 31 k/icd)
You'll net a fleet that could look like this:
4 CAs + 10 CLs (31.6 K/icd) with 127 sea attack in the fleet (higher than the BB fleet and CL monoculture fleet)
OR
5 Cas, + 8 CLs (30.1 k/icd SAVINGS!!) with 125 sea attack in the fleet (higher than the BB fleet).
The *only* consideration here is that the techs required for CAs and CLs are in different trees. But the techs for CAs are cheaper than either BCs or BBs to research themselves, so you could even more reasonably cheat on them and research ahead of time. You will likely lose CAs in a SAG battle but they're much cheaper to replace than if you were to lose a BB or BC.
The craziness gets more pronounced if you look at building multiple fleets. For example adding another BB fleet gives you a total of 6 BBs and 14 DDs, but for the same price you could field 8-10 CAs, and 16-20 CLs. With a sea attack comparison of 219 vs 254/250
With armor you have a few things to consider.
1stly, you could build *some* AT - AT is pretty cheap over all (it's the cheapest arty unit there is) and it provides *some* soft attack - but for Japan, especially if you go the carrier route, CAS is far more cost effective against tanks than building your own tanks, TDs, or AT brigades. In part because your AT brigades will hurt you in amphibious operations and in poor INFRA environments (because they require more supplies than another INF "brigade" or regiment). But they also provide the direct fire combined arms bonus so that's nice - especially if you want to set up a combined arms OOB - may as well attach AT to some INF units to train the generals in combined arms. The other upside to AT is they never go out of usefulness - as long as you keep the AT tech updated they'll still be effective against armor.
CAS is generally the preferred way to go for me, if you build carriers, because the CAS doctrines make your CAGs better, your light air craft practical will be high anyways, and you can move the CAS around much more effectively than an AT gun toting division. With the exception of the invasion of the USA - any time you encounter armor they'll be in such a crappy INFRA environment that it will be almost impossible to keep them in supply, so you won't have to worry about them for long

(India/USSR). Before you land in America you're going to want to build at least *some* AT because the US typically has oodles of armored stuff - especially if you wait until later in the game to invade.
As far as SH:BB goes, I don't really like them for Japan, because you have the LS to burn on upgrading your stuff normally. SH:BB make sense for nations with IC to burn but little LS (like a Chinese Faction) or the USSR where you may not have much LS and hardly any naval practical/theory. So the cheapest (research-wise) route to building a modernish ocean going fleet is SH:BB + DD.
Let's take our 1940 example for a war with the USA again.
For the price of the 3 BB + 7 DD fleet (31.8 k/icd), we can build:
3 SH:BBs + 4 DDs (31.9 K/icd) speed: 27.7 (too slow to catch a 1940 carrier) with 102 sea attack (only better than a pure DD fleet in the same price range)
OR
2 SH:BBs + 10 DDs (32.25 k/icd) speed: 32.2 (fast enough!) sea attack: 101 (still less than any other fleet than pure DD).
If you for some reason didn't want any kind of a SAG, and wanted to spend the absolute minimum of research on it, as Japan, building *some* SH:BB could be used for coastal protection, but that shouldn't be necessary given the huge amount of ships you start the game with.