We bring down our dog 4-5 times each year
Yes there is a pet passport which is valid for 1 year it costs $100. You will need to provide a US vet certificate of health. Once you have the passport you just pay a BZ 20 "inspection fee" each time you arrive and you need to have a current US certificate of health.
Jerlyn Tucker at BAHA is always helpful getting our pet passport. We arrived yesterday and our passport was waiting at BAHA for us.
If you contact Jerlyn she can E mail you a application form or E mail me at [email protected] and I will send you a copy

Colin, Marianne & Sophie the Dog