I can’t tell you what to do. I can tell you that your boyfriend’s behavior is unreasonable. It doesn’t look like he has recovered from his prior break-up at all. Sadly, the things he is doing in an effort to prevent you from abandoning him will likely make you do just that. I doubt very much that you can sign on for a lifetime of being distrusted like this.
I do believe that people can change. But they have to want to and they have to want to put in the effort. It doesn’t sound like your boyfriend thinks he needs to change at all. Instead, he wants you to do the changing when you have nothing to change.
It’s worth a heart-to-heart conversation. Suggest that he get himself into therapy to work through his trust issues. If he won’t do it or if he only goes for a few sessions and quits, you have some hard decisions to make. If he agrees that he’s got a problem and puts some real effort into it, your relationship might have a chance.
I wish you well.