If the dog was 'largish' and not in a carrier, it must have been a service dog or "emotional support animal" with correct paperwork approved by AA prior to the flight.
As AA is legally obligated to transport these animals accompanying their handlers/owners in the cabin, you basically have no recourse. They cannot make the animals' handler buy a second seat, and when push comes to shove (seatmate with allergies), it's the animal that flies.
BTW, this situation could happen on any US carrier who are all bound by the same regulations. In fact, AA Special Assistance is tougher than others when it comes to approving ESANs/ SVANs.
Of course you can try writing in to AA Customer Relations (use the
form on the website), and they might throw a few miles your way, but that's about it.