You already know the answer.

You either break up or you accept the abuse.

You are not the problem. She is. She has major trust issues and jealousy issues. There are probably lots of insecurities also.

That is not something that you can fix. And it's not something that she can fix on her own. She needs to go see a therapist to get over all her issues before she can have a healthy relationship.

There is absolutely nothing you can do to make her not think that you are cheating. There is absolutely no way to prove that you are not cheating.

You will not fix her. Talking from experience with this kind of girl.
It only gets uglier.