I can't speak to AA, but with SN, I have forced a higher booking class by booking dates when the cheapest available booking class was the one I wanted (using ITA Matrix's calendar search and looking for more expensive dates), and then simply rebooking the ticket for free for the actual dates I wanted - that would keep the higher booking class, and they have got rid of rebooking fees (or at least that was the case during covid). No idea if that would work with AA though.
Otherwise, I guess you could call...