Having just cleared into the Turks & Caicos with one dog and one parrot we can tell you what's needed. If you are already in Bimini you'll need a veterinary certificate from a Bahamian vet - don't expect a US one to work as they look for clearance from the "exporting" country. If you did everything you were required to do to get the animal(s) into the Bahamas
then it should be a simple process (as long as all shots and rabies titers are current) of visiting a local vet, or have them come to the boat (typically about $150) and give you a vetinary health certificate. If you have a CITES listed pet as we do - it's a little more complicated.
Every country has different laws, but pretty universally they don't care where the animals
originated, but where you're NOW comming from, so if for example you go from Turks & Caicos to the Dominican Republic
you should get another vet certificate from the Turks & Caicos, or even if you're just going back to the Bahamas and have been in T&C - though we're informed the Bahamas will often permit
re-entry within 30 days of the issuence of the original health certificate this is not guarenteed.
Turks & Caicos are beautiful by the way - well worth the trip. Recommend Southside marina if not anchoring
out, better rates and very friendly, however others are closer to downtown.