I would say if there was a fair amount of land around the property it should be easier.
Such as a half acre.
Dogs are risk, and landlords don't want risk.
The stigma with pit bulls is not a good one.
If you ever called around for insurance, you would find it is almost a no win situation.
As it stands in California there are few insurance companies who want to insure a home that is rented out.
Couple that with Pit Bulls and you may need to rethink renting and perhaps check for zoning and perhaps try to buy a home at a reasonable price that fits your budget. Keep in mind you will have to have homeowners insurance when you buy a home.
Harold Sharpe - Broker
So Cal Homes Realty
California Department of Real Estate License # 01312992
All the best
Dave & Lisa