So you're cruising the Caribbean. Are you obliged to take your cats through immigration every time you arrive a new island nation?
Most islands will ask you to declare your cat as you clear in. If you anchor out as I do then there generally there are no further formalities or fees , except Antigua where you have to pay for the Ministry vet to visit and perform a health check. $50 US the cat may not need to be present.
This is my 13th year cruising with cat[s] and no problems. Having some sort of inoculation record may help but I have only been asked for it once.
NB it must have been blowing 20 knots + as I carried full sail to 20 on that boat.