DOT approval allows transportation of cylinders by common carrier and, depending on the contents, even when fully charged.
However, the airlines (and the Captain of any flight) have the final say as to what they will carry.
Some airlines will allow an empty tank to be shipped on the flight. Others require the valve be removed. Others won't allow them at all.
Even then, the Captain of the flight can refuse to carry cargo he or she feels compromises the safety of the flight. Remember, the Captain is ultimately responsible for the safety of the passengers and crew.
I do wonder why you wanted to carry a tank at all. All dive shops in the Caribbean have tanks to rent, many are included in the package price. Plus you have to have the tank visually reinspected upon arrival on the island and again upon return to the States. Not to mention the possibility of the tank gettign delayed because of weight restrictions. Finally, if the tank is not in a secure container, you could be held responsible for any damage to the airplane or contents shluld your tank get loose.
If you ABSOLUTELY must have your own tank you might check into shipping it to the island. If you can't do that, plan on some unabashed begging of airline personell.