I imagine you can use your AA number up until just before you leave the lounge to go board, then have an agent change to your BA number. You won't get priority boarding (unless you still have/show the old boarding pass), but that's probably the least valuable perk, particularly on a long-haul.
I guess there's an outstanding question of whether any preferred seat assignments would then evaporate, but I suspect that's not something that would happen automatically (could cause all sorts of problems if it did right before boarding!).