I think I remember having success by saying at booking time that I was booking for someone else, not myself. (Of course, you can redeem BA points for someone other than yourself.) Then input your own name and other details but don't give input a frequent flyer number. Then you have an AA reservation with no frequent flyer number and can attach your AAdvantage number after booking.
Of course, that doesn't work once you've already booked the trip. And it's been multiple years (pre-pandemic) since I've redeemed any miles.