We flew Continental from Minneapolis to Belize with a change-over in Houston. Continental is actually a pretty annoying airline. For example, my husband and I booked seats together and when we went to check in they moved us apart and charged us $69 to move seats together again (I got the money returned once we got to the airport but it was really annoying). But, they do guarantee your pet can get on the plane (some airlines I think have a limit of numbers of pets on the plane and it's first come first served). American Airlines does not let pets fly in the cabin to Belize. The two important lessons I learned were: when getting the permit from BAHA to bring the pet in, you can fax or email the information but they prefer fax. We emailed and although the BAHA office person emailed the permit to the inspection guy at the airport, he doesn't check his email so there was a little delay and issue with that when we arrived. Yeah: fax your BAHA paperwork (all the forms and instructions are on the BAHA website (http://www.baha.bz/
Also, you need to get a vet certificate before you bring the pet in to Belize. We didn't realize until we went to the vet that the US (if that's where you are) also requires a signature from Dept of Agriculture. There is an office near us and we were able to run over and get the signature but it would have been good to know that in advance and be able to plan it.
We bought a Sherpa brand pet carrier on line from PetSmart. It's pre-approved by most airlines for in-cabin travel and there were no issues at all with the airline or airport security at any of our stops. We had a long layover in Houston on the way back but they have a pet exercise area so that was perfect. My dog is about 12 - 13 pounds (13 on the way to Belize and more like 12 on the way home after all his vacation action!) and he was comfortable under the seat. I'm not sure you would want to take a dog or cat that was much bigger than that.
The BAHA officer at the Belize City airport was great!