I don't know of any places to rent on AC but my tip about bringing pets in to the country is that it's hard to do the whole thing by email (especially initial contact). It's worth the investment of an international phone call to ring up BAHA and talk to Melody about the process. She'll walk you right through it.
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Make friends with Melody. She's the ticket.
We were actually able to do it entirely through email and fax, no phone call needed. Ms. Melody Robateau at BAHA is super helpful (email: animalhealth.baha@gmail.com).
Like everyone said, don't rely on the BAHA agent at the airport having a copy of your paperwork. I put a copy of each item in a page protector and into a small notebook, which I brought with me on the plane and was able to show it all to the BAHA agent (who indeed did not seem to have my paperwork).
It's a bit harder to find a rental if you have a large dog (versus a small one) but certainly very possible. I'd start with B-Lease Mgmt (//ambergriscaye.com/blease/).
Good luck!
Emily
P.S. Edited to add: I had not read all the replies before I posted the above -- my bad. When we bought a condo north of the bridge in 2010, we found that only a handful of condo complexes on the island allow pets (at least at that time). And that's for owners. Even fewer will allow pets for renters. That said, there are plenty of houses that do allow a renter to have pets, and a few condo complexes. Reef Village does as does Royal Palms. Hope you find something!
AmbergrisCaye.com 
