Hi Amelia516, welcome to the group

You're in the right place if you're wondering if your spouse may have BPD. The cycle of rage and accusation, to niceness and rationality, and back to rage, is familiar to so many members here.
It sounds like you've already tried the approach of: "If I can just prove to him that he's worthy of love, he'll realize it on his own, and things might get better" -- and it didn't work. That's one of the difficult and sad things about BPD -- for the partners to change, they have to want to do the inner work on their own, and because deep shame is a part of BPD, it is challenging for pwBPD (persons with BPD) to stick to effective therapy. Not impossible, just difficult.
Fortunately, even though we can't control what a partner does or doesn't do, there are things we can do and learn -- nonintuitive tools and skills -- that can start making things "less worse" and that can "stop the bleeding" as is often said around here.
These tools and skills aren't a magic wand where the relationship will then be completely wonderful and just like a "normal" relationship. Often, relationships with pwBPD are described as "emotional special needs" relationships. Just like we wouldn't expect a person using a wheelchair to be able to walk up stairs, we also need to recalibrate our thoughts about what pwBPD are emotionally capable of. Once we accept those limitations, there can sometimes be more peace in the relationship.
Anyway, all that to say -- you did include a perfect amount of information

Always feel free to share at a level that's comfortable to you. As you get settled in here, I'd invite you to check out our section of articles on
"When a partner/spouse has BPD". See if any of that resonates with you.
One question before I wrap up: how old are your kids? How have they been doing with the conflict?
Let us know how you're doing, whenever works for you, and again, welcome;
-kells76