Bismarck had no horrible AA, at least for its time (1941, compare contemporary layouts and you will find that it is more than competetive to other BB).
Prinz Eugen had a decent AA as well (it repeatedly shot down soviet anti shipping planes in 1945), however AA is limited as in all warships. Do do not exaggerate the US 5"/38 DP AA gun. As a gun it is way worser than PE´s 4.1"/65 in terms of muzzle velocity, platform, range, effective ceiling and trajectory. The VT-fuze was what made the 5" effective, not the gun itself.
The Baltimore class, unlike Prinz Eugen, has grave shortcomings in underwater protection.
For Bismarck please shift to ww2general, there is a thread covering a good deal of discussion.
The question is -what is important for a BB?
1.) It has to stay in the battleline and keep it´s vitals safe for some degree
2.) It has to deliver a good deal of damage
Yamato gets all of it. (The sheer number of planes involved in sinking Yamato&Musashi would sink any Iowa-class BB as well and certain points of it´s design (metacentric and byuoncy-reserve, unprotected sides, shortcomings in TDS) makes me think that Iowa would sink pretty soon under these circumstances. AA capability is neglectable. I found out that NOT EVEN IOWAS REPUTATED AA HAS ANY CHANCE TO PREVENT AN INCOMING FLIGHT (12 planes) FROM REACHING BOMB DROPPING DISTANCE ON IT´S OWN!
And BC?
1.) Speed in order to dictate the terms of engagement
2.) Range
3.) a decent firepower
4.) a good protection
As far as I see, Hood covers most of this (at least in the proposed refit), Alaska suffers in protection (and I mean it suffers a lot) and Scharnhorst suffers in firepower (at least sufficiant for a raider but useless against enemy BB)