A suggestion, get to a computer and find one of those sites that have you, "Pick the Pit bull" They are all a little different, but basically, they show several photos of pure bred dogs and then you have to pick what one is the Pit bull.
Print the images and the answers.
Find a vet who will write you a letter saying your dog is some other breed. Maybe even re license the dog, as the other breed.
I know it is "shiesty" and dishonest. But consider this, go to any shelter, do it online, or use like adopt a pet or something. So many of the dogs for adoption are called breeds that they do not even look any close to.
If you adopted your dog, chances are, it is not the breed you think. (wink)
Heck, print a few pages of dogs that are clearly not the breed the shelter is claiming.
Many places restrict based on size, instead of breed. But if you have enough money, a big deposit will get you into some places. You could also get renters insurance coverage for your dog and let the new landlord know you carry it. It's not very expensive.