Re-read the AA page since it doesn't say no pets to the DR in baggage or the cabin at all. It even has the rates for checked and in cabin for the Caribbean. They prohibit birds from the DR TO the US.
Notes:
Dominican Republic
Birds from the Dominican Republic are prohibited entry into the United States.
Transatlantic, Transpacific and Central/South American flights
Carry-on pets are not allowed on Transatlantic, Transpacific or Central/South American flights. Animals traveling as checked baggage may be allowed, however, some international destinations may limit the number of animals allowed per passenger as checked baggage. Please call American Airlines Reservations at 800-433-7300 and check with the consulate of the country to which you are traveling for further details as the answer varies from country to country.
American Eagle does not accept pets as checked baggage for Priority Parcel Service between San Juan, (SJU) and St. Kitts (SKB) and between San Juan, (SJU) and Nevis (NEV).
Fees:
Checked Pets
Within and between the United States and Canada, Mexico, Central and South America and the Caribbean
$100 USD / $137 CAD* per container
Pets Traveling in the Cabin
Within the 50 United States
Between the 50 United States/Canada/Puerto Rico/USVI
Between the 50 United States/Canada and Mexico/Caribbean
$80 USD / $109 CAD* per container
Look here:
http://www.aa.com/content/travelInformation/specialAssistance/travelingWithPets.jhtml