We moved here with a Saint Bernard from Canada, We found it easier for us to come down thru Cancun and our dog flew Continential into Belize City
You will need up to date shots and a certificate of health, and you will likely have to pay some duty on your dog (It was 206 american dollars total for us)
Continential after much research seemed to be the best and safest bet for an airline (tho if your dog is small you may have other options).
When you know the date you or you and your dog are arriving you should also contact BAHA - Belize Agriculture Health Authority - as they will make sure someone is around to facilitate your passing thru customs easier for you and your pet.

If we give Peas a chance won't the Lima Beans feel left out ?