I think the question of ‘whether BBs are balanced or not’ is moot if they aren’t purchased in competitive play.
As a brief aside, I think IL is partially correct in that if Cruisers become truly viable, it takes a bit of pressure off of BBs needing to be less expensive. Why? With an affordable naval punch + amphibious support unit like a cruiser finally available, BBs are free, finally, to assume the one niche that their sculpts and stats indicate them to have: the ponderous King of the Seas (or, at least, of the surface). In this apex role, they should be expensive! Leave BBs at 20. Hell, make them MORE expensive; 22! But make them fearsome in their role: they should be unparalleled at providing raw naval dominance, and should, IPC for IPC, outslug the other surface vessels (when escorted).
I see three easy options for the Battleship, three less easy options, and one that is totally different:
EASY
1. Add dice (two attacks, or a one-time/recurring AA roll).
2. Make existing dice stronger: make it 5-5, or roll twice and take best result, etc.
3. Reduce the cost until they at least compete with a pure DD fleet of equal value.
LESS EASY
4. Require, if the BB’s owner declares such before rolling, that a BB’s hits on a given round be assigned to surface warships only
5. Allow them to give/receive a combined arms bonus to/from other warships
6. Do away with the 2-hit mechanic, and instead let them soak one hit every round- as in, ignore it completely.
SOMETHING TOTALLY DIFFERENT
7. Let them buff (+1/+1) the surface fleets they succor.
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Option A (strong): “All of a power’s non-BB surface ships (DDs, CAs, and CVs) in a Sea Zone attack and defend at +1/+1, as long as at least one of that power’s BBs is also present in that SZ.”
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Option B (weak): “For each of a power’s BBs present in a SZ, up to one of each type of that power’s other surface ships (i.e., 1 DD, 1 CA, and 1 CV) attacks and defends at +1/+1”
The more I think about it thematically, and the more scenarios I run, the more I like option 7 (let’s call it “Flagship” or “Command”). Run some scenarios yourself: it is easy to model in calculators. Setup your fleets, then convert the desired number of BB-escorting DDs -> CAs, CAs -> damaged BBs, and CVs -> DD/CA (depending on atk/def). I think that this sort of command mechanic just perfectly corresponds to what a BB physically is in Axis and Allies: A huge hunk of lovely, menacing plastic that looks like a game-changing capital ship, if one can juuust afford to put it on the board; yet unwieldy and vulnerable when left alone or overrun.
I do also like option 4 (call it “Barrage”), not just for flavor but help a fleet with a BB reliably bypass at least one point of Submarine shielding each round, because subs are currently so strong.