My wife and I twice flew our toy poodle in from New York via Charlotte on USAIR. She had to ride in cargo since USAIR's policy doesn't allow pets on international flights. It was the beginning of August and I don't recall the outside temp, but she was only on the plane for 5min before they took her off. Belize (BZE) is a very small airport.
I do recommend that you try to plan on a long layover going back. These days they give you about an hour for you and your bags to get through immigration/customs and that is not enough.
We had an almost 3hr layover going back to NY. We were able to retrieve Peanut from the baggage claim and we even walked her outside of the airport. The next part is funny, because we had taken possesion of her and her kennel, they had to get a TSA employee down from the office to frisk our 6lb dog and her kennel for bombs and box cutters. I understand completely, but it was a funny scene.
My best advice is to HAVE ALL THE PAPERWORK going both ways and just relax. The airlines basically know what they are doing, you pay extra to bring your pet for a reason.
Also if you are coming to the Cayes, I recommend the flight from BZE rather than the water taxi. Just what your pet needs is another high-speed, bumpy, traumatic travel experience right after the long flight(s).