We just brought a dog down on Thursday. Dogs in the cabin can be cooked online - dogs going as cargo need reservations. Here are the U.S. numbers 800.575.3335 or 832.235.1541.
Last year I called Continental direct in Belize City and made the reservation. Here are the numbers: 0800.266.3822
501.822.1062. These are local numbers.
You need a certificate from a vet certifying that the dog is healthy (Saga can do the physical) and that it has its rabies shots, an appropriately sized crate with a food dish and water bottle attached. You then take the dog with you in their crate to the same counter you check in at in Belize City.
Hope this helps.