Hi Rick and Cheri,
I realize you guys may not want a car in Belize, as you'll be living on Ambergris, but you know, the drive is beautiful, and it really only takes 3 days from the American border. We took our two dogs and two cats, and actually found several hotels that would take us with the pets (it helped that it seemed to be off-season). We drove with them both there and back, and had no problems whatsoever. The only grief is all the paperwork you have to do for the vehicle at the borders.
I know our one dog would not be happy with the plane travel, as she has enough issues with separation anxiety already.
Just a thought, and hope you have a good trip whichever way you can get things to work.