My odds calculator tells me that there is about a 25% chance of the bomber winning, 25% chance of the BB winning and 50% chance they both die. This assumes one-hit BB and no heavy bombers
Considering that while the BB costs more IPCs than the bomber, keep in mind it can do potentially much less attack damage (one time potential hit only on amphibious assaults) and on defense is essentially equal to only two transports (my odds calculators says 45% BB, 45% transports, 5% both die) I wouldn’t worry about it. Let them bomb away and your opponent will likely lose their bomber.
However, if you really don’t want to lose the BB, just make sure you keep a transport or two with it for “fodder”.
Its usually not a good idea to split up your navy anyway, I always make sure to keep some transport or submarine fodder next to the powerful and expensive ships whenever possible. As for land operations, you should always keep cheap (infantry or transports) weapons alongside expensive items (BB or bombers) so if they are attacked, the cheap items can be eliminated first.