Personally, I don't need my partners or partners' partners to be best friends. It would be lovely, don't get me wrong, but I think that is unrealistic. Having said that, I need people to be able to meet and be civil to each other. If they can't manage that, that is a deal breaker for me. I wouldn't be happy in a situation where my partner's partner did not want to meet me or get to know me.
It sounds like you have a really bad vibe off this woman, and part of me is wondering why your partner isn't listening to your concerns. The information you have given so far about this woman suggests you don't trust her to be respectful of your relationship, and that would raise alarm bells for me.
Hope the situation gets resolved xx