Deb, I took my cat last year to Bahamas
...never declared him, never got any Bahamas paperwork for him. No one ever asked. Rarely does anyone actually board your boat when you check in to the country, so don't fear that someone will find him during an inspection
My cat has all his papers, and if I ever got asked, I was going to say that the permit
was applied for. All the cruisers I talked to with cats said they never bothered with any paperwork. This assumes your cat will stay on board. Not sure what would happen if you had to take the cat to a vet over there.
I think they are more worried about dogs
that would go ashore, risking a spread of deceases; pets
that stay on your boat don't seem to be an issue.