I would also make sure you have up-to-date documentation about rabies shots. Also is your pet spayed or neutered? If it is you should document that since behavior is drastically different in intact canines. The problem is that there is a negative public perception about the breed, deserved or not, and so in a competitive rental market your application will be viewed as less strong because you come with a pitbull. There is no way around that if you keep your pet.
In this area there have been some high profile dog attacks in the last few years so you are fighting against some strangers perception of the breed. Don't do what some people do and start hiding the fact by calling it a Stratford Terrier, etc. because that will annoy people when they find out it is a pit.
I would ask landlords their position on the dog first before even seeing their rental to save you time.
Out of the box idea: If you trained your dog to be a service dog you would have a possible solution since then your dog may have to be allowed....
For the long term, this is a good reason to figure out how to buy a home of your own. You can buy a large condo and mortgage payments are equal to what you would pay in rent. It is an easy process when you have an entire experienced team walking you through it holding your hand. The process starts with a free, no -obligation phone call to a mortgage broker (call me and I'll refer you to a reputable broker) so s/he can tell you what you need to do to get yourself into position to buy.
For the short term, get 4 or 5 reference letter from people who know your dog well. They need to be convincing and have some detail with examples of how your dog is calm and well trained. One strategy is to find a place you want, and as long as they don't specify they absolutely will not rent to dogs, make a superb first impression on the landlord. When you are offered the place, then produce your dog's letters of reference. Wait until the landlord attaches to you before you tell them about your dog. Offer to buy your own insurance for your dog, too. It is inexpensive.
If you need other help, give me an email or a call at: Hilary@ilovesonomacounty.com or (707) 545-6555
Keller Williams, Realtor