Another thought that may help you. United flies from Houston, TX so I don't know if you can make a connection with them but you could rent a car if nothing else and drive from Alabama to Houston. Call United "Pet Safe" or Pet services, it is a separate number and talk to them.
Your dog will love it here. Mine has been coming here for winters for 7 or 8 years and he is in heaven while here. He runs and gets in his kennel when I get it out. The other thing is that on the Belize end if you are coming out to San Pedro make arrangements with Tropic Ferry because it is the least stressful way to get out here with a pet. Good luck and if you need more help just pm or ask on the message board.