One thing you can do, I've seen people do it is....
There docs will write what is like a prescription to let you have the dog. Sort of like a service dog. I'm not sure how it all works, but people get it done with just about any breed of dog. It doesn't have to be one that will fetch slippers, get the phone, open doors or what not....it's a dog that's more like a companion sort of thing.
Might be something worth checking into. Once it's said that the dog is more like a service dog, landlords CANNOT not rent to you. It's a violation of the Disability Act.