Future brother in laws girlfriend is a bitch and don’t want her at the wedding

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My fiancé and I have been together for 3 years and just recently got engaged. My fiancé has a brother that he’s really close with but his girlfriend of 6 years is bitter towards me. She comes off as if we’re in a competition and she’s constantly making petty remarks. She’s a few years younger and doesn’t have her life together as much as me and my fiancé do but isn’t doing bad at all. We’ve had multiple discussions about her behavior towards me after I’ve called her out about it. We always end up “making up” but the next time I see her she’s back to her ways. My fiancé’s family is very close and they’re also aware of how she is and treats me. My future mother in law even mentioned to me after I said how I try with her that if she was in my position she would be done with her and frankly that’s where I’m at with her as well. So my question is do I have to invite this girl to my wedding? I get along very well with her boyfriend (my fiancé brother) and I would hate to see him being upset bc I know it would also upset my fiancé but I really don’t want a person at our wedding that is consistently rude to me but also don’t want to cause a divide in the family.