First of all, I'm sorry to hear you're dealing with a person like this. It reminds me of a guy who was on my previous team. Relatively competent. Stuck at Senior. Had a lot of issue seeing (often younger - myself included) engineers getting promoted to Staff, Principal, etc. over him. Knew that he had an incredibly toxic and abrasive personality and thought merits alone should allow him to move up the ranks. He would reign it in sometimes after people complained either to him or to the manager, but he'd go right back to his old ways a few days later. So. Frustrating.
Anyways, my advice is: document. Document specific instances of behavior that you see that is toxic or disruptive to the team.
It sounds like this person needs to go - or someone else needs to start addressing their behavior with them. Either scenario will require that person (your manager, skip level, or HR - or some combination) to have examples of specific instances of behavior that needs to be addressed.
Others have recommended the empathy approach. Talk to the guy. Take them out to lunch/drinks/whatever. If you haven't tried it at least once, I agree it's worth trying. There's always a chance your reading things wrong or don't have the whole picture. But if you have tried kindness and understanding, and things don't change or they only change temporarily... I don't think the "kill them with kindness" approach will work in this case. It's the approach I tried in my situation and all it did was drain me and take away all of my excitement to work on the project he worked on. Because this person will never change and you can't make them change.
Whenever I've been around toxic people like this, I always use the "if everyone's a jerk, you're the jerk" heuristic. Because I certainly can lose my cool and be a jerk sometimes. Not always intended. So in order to ground myself, I ask: OK... who do I have problems with? Just one person? THAT person? Yeah. Anyone else? No... everyone else is pretty awesome! OK... what about them, have they expressed they have problems with more than just me? All but one person on the team!? Huh. Yeah... That seems off. It's far more likely they're the jerk.
Certainly not a hard and fast rule. But it's helped me stay sane through abusive relationships both professionally and personally.
Last point - unless this person reports directly to you, they're not your responsibility to fix. If you've talked with them and expressed frustration with the _behavior_ you're observing, you've already done your part. Your management needs to step up. If they disagree with the state of reality, then figure out why there's a disconnect. That's where documentation is your friend! If you can convince another coworker to do the same and they also can provide documentation that corroborates or supplements, even better. I wouldn't be too afraid of being clear to your manager that this is something that you need fixed in order to feel comfortable and productive at work. It doesn't need to be "it's me or him". Make it about how _you_ need help from your manager. If you can show them that effort has been made on your own to address the situation and it's not working, then any competent manager should work with you both to find some sort of path out of this. If they continue to refuse (especially if multiple team members are expressing the same thing) then I would consider escalating to your skip level and then to HR. Chances are, assuming this person really is a jerk, that these people are already aware of what's going on. If no one seems to care that a bad personality is dragging the team down, then that's a huge red flag on the company culture as a whole and I would gtfo as quickly as I could line a new job up.
Good luck. I hope you can find some relief here. Life is too short to put up with other people's crap.