Don't
That is my one word.
I have a very loving cat on board. He is both a blessing and a curse.
Concerns:
1. I have been in about 16 or 17 countries in the last 2 years. With the exception of three they ignored my cat after I assured them that he never leaves the boat. I doubt that they would accept a similar plea for a dog. In the 3 countries that did require me to clear in my cat it cost me more to clear in the cat then it cost me to clear in the boat and all the crew combined. It also took substantially longer to clear in the cat.
2. You are bound to your animal. If you want to take a several day "shore leave" - perhaps to visit a nearby city or attraction you need to find someone to take care of your animal. In most countries your pet is not going to be welcome on the bus, train, cab, rental car, hotel, etc. Although sometimes you can find someone to pet sit for free in my experience you generally end up paying something - even if it is a thank you gift for the free sitter.
3. Disease: just like you will find that the local "bugs" can get you down until your body adjusts the same is true for your pet.
4. Cleanliness: I never had a problem with fleas etc. while cruising with a pet in the U.S. In a lot of other countries their standards of cleanliness are much lower - we had rats and cockroaches in the marina at Guadeloupe (this was the high end marina!) and it seemed to be accepted. XO (my cat) got fleas while in Guadeloupe and they didn't die as easily from the US flea medicine as I would have hoped. BTW that is also true of the boat - a bug free boat can quickly become a haven for bugs as you start to cruise if you are not diligent.
As Shakespeare would say, an aside: My description above includes just about everywhere on your list - Mexico (the East side) Belize, Guatemala, about 12 Caribbean Islands, BVI, Turks, Bahamas....
I hope this helps ..
Fair winds and following seas: