There is last minute and there is last minute

BA redemption tickets have to be bought in advance of 24 hours before the flight although a call centre can consider taking a rewards booking if less than 24 hours. Given this is an AA flight, I doubt if the call centre has any discretion to make a call.

You need to see if there is an AA has any AA saver tickets as oppose to Anytime tickets. BA allocation is based on AA saver tickets only.

This is more complicated since I suspect you will have to change meaning checking if there are AA saver tickets on two segments rather than just one.