It's difficult because we can't see this dog in person or witness your interactions. Your descriptions sound like a lot of behaviors I'd expect from a high-energy, intense bully breed - low impulse control with play, getting too mouthy, jumping. You're right that these are inappropriate behaviors that need to be worked on, but they're not behavior problems in the sense that they don't sound extreme, abnormal, or dangerous (in the sense that they indicate he will attack you or another animal, obviously jumping and climbing on people or getting too mouthy can cause injury).
Maybe I'm getting the wrong impression because, again, I'm not seeing him or these behaviors, but it's hard to offer advice for something when my impression is that some foundation training and/or lifestyle adjustments would improve things, but the person living with the dog is saying the animal might be dangerous. That makes me reluctant to give advice along the lines of "try this training protocol" in case I'm wrong and the dog does have a more serious issue that needs special handling, which is why I try to suggest in-person trainers in that situation, or a new trainer if the one you're using hasn't been able to help you effectively address the issues.
I do wish you luck, and I hope BFAS helps you work out the best plan for your and your dog. Again, you did an awesome thing taking him in, and it's obvious you care for him.