Since I have checked coolers in the past, I had to look this up and you are correct. The AA website states that boxes are prohibited as checked luggage when traveling to the Caribbean and defines boxes as:
*A “box” is any container that isn’t normally used for transporting items for air travel; this includes plastic tubs, containers and coolers.
Although I don't understand why AA objects as long as it meets their dimension requirements, I wouldn't want to lose a good cooler. Maybe your charter company can come up with a spare.
Fred