Is this the dog in your signature? He definitely has a fearful look in his eyes... I've met just a few Pit Bulls who were that way (out of COUNTLESS over the top happy Pit Bulls). Very nervy and quick to become defensive, and they had really nice owners. One was a bait dog, completely earless and scarred up, who showed that she was a really sweet dog after her very dedicated owner spent years working with her. I also know a wonderful, amazing Pit Bull owner who shows/works her dogs and has several rescues - including a little white girl, who is scarred from fighting and mange, but super cute. However, that dog is a walking timebomb. She took a chunk out of my co-worker's thigh for letting her guard down around her, she was walking to the other side of the playroom! That dog scares me, truly, but her owner is dedicated to managing her for life and I respect that and trust her to.
I'm glad you're seeing a behaviorist! No one here can tell you what to do with your dog, especially without ever meeting him. I will say this - if there is any question that you can't keep your dog fully contained and keep the public completely safe while undergoing the training process, PERSONALLY I would PTS to protect the public and the breed in general. Leaving him in a backyard unsupervised is not acceptable in my opinion. He should be kept from any strangers, watched with a hawk eye, and handled with absolute care and concern. NO pushy methods, all calm, non-confrontational, positive, and move at a snail's pace with him.
I couldn't handle it if one of my dogs bit me.
I don't do anything to them that warrants a bite, but even if I did, I'd expect them to be forgiving. It's very sad when some one's dog bites them and concerning depending on what you were doing to receive the bite.
Honestly Gonzo was like that when I adopted him, he would definitely bite a stranger if they grabbed his collar. He bit a neighbor for no reason at all. He has come a very long way to being a well adjusted, confident dog who is safe with people. Anyone coming over used to be a massive ordeal. Now it's nothing! I highly recommend reading Click to Calm, that really really changed the way I viewed his fear aggression and it made me realize he COULD be a trustworthy dog if I just showed him that the world wasn't scary. I do believe that Gonzo is not well bred and has terrible nerves, but breeding isn't everything.