I do not recall any Caribbean island that restricts dogs or cats, but you should have, as mentioned, all veterinarian records for your animal, with documentation of it having received all it shots. *Central and South America also do not, to my knowledge, prohibit pets on board. *You can usually check each country's web site to learn more about their quarantine restrictions. *Another information source is noonsite. *
Cats may be more self-sufficient regarding staying on the boat, but they are better able to sneak off the boat when/if you are tied up at a marina. *For this reason you will probably not be allowed to tie up in a marina in most of the South Pacific islands or in Australia and New Zealand if there is a cat on board. *Although South Pacific islands will not be as diligent about checking your boat to be sure your animal is still on board, this will be done regularly in Australia and New Zealand at a significant cost to the boat.
Having a cat on board can be a benefit by discouraging vermin from coming aboard, which they can do. *
One final comment. *The tropics is full of lots of parasites that do not survive in the temperate regions. *Be vigilant about the possibility of your animal contracting worms and in Trinidad and Venezuela be wary of rabies-infected bats.