I'm actually quite curious as to why the battleship does nothing to air units? I know it would make this ship OP as heck, but let's face it. During the time, they were massively OP.
now I'm not suggesting a massive armament of .50 cal and bofors but the original design for the North Carolina class Battleship included a small amount of 40mm bofors and .50 cal and was increased largely during the war. Why not implement the original AA defense but make it weak, obviously, but give the Battleship a chance to defend herself if the battlegroup died.