Most places in the Caribbean are just fine as far as your airplane not having all the avionics stolen out of it if it's sitting there for a week. Expect landing fees and expensive fuel, and they have no shortage of forms and approvals. Careful with the pets, it's not going, it's coming back. US just changed its rules on domestic animals entering the country (I'd have to google it for specifics) which caused a lot of snowbirds to the islands to leave their pets at home.
Recommendations? Depends on how far you want to go. Not Jamaica and probably not DR. Puerto Rico has GA-friendly airports and it's the US.