I'm wondering whether, for the longer term, one answer might be to ensure that the F&F list has "another" person of the same name as the OP who belongs to AAdvantage rather than to BAEC, and for whom the OP makes the booking whenever this situation arises. That way, the booking will not have the OP's BAEC number attached but the "other" person's AA number instead.

Of course, if the HHA really isn't needed, then there would seem to be few downsides to breaking it up.