American Airlines does not allow pets to be checked as cargo on their Airbus narrowbodies. They are growing this fleet with A321neos that begin delivering at the start of 2019.
Airlines are cracking down on emotional support animals. Small pets can travel in the cabin for a fee, but larger pets need to be checked. Owners of larger animals then may be caught between Scylla and Charybdis unable to bring their animal on board as an emotional support animal, unable to bring them in cabin for a fee, and unable to check them when traveling on half the domestic fleet — and many destinations are served exclusively by these aircraft.