I just booked booked LHR-DFW-SAN-DFW-LHR in F via BA Holidays
included the hotel rather than a car because I am going to the hotel anyway
AA domestic sectors showed as "Business class" while making the booking
But I have just gone to AA.com and selected seats in the domestic F cabin
Not sure what would have happened if I had proceeded with the original booking on the main site, booking flight then adding car.
But the Holidays part of the website seems to work correctly.