Have to invite my inappropriate friend to my birthday

Hey, so my birthday is coming up soon, and I'm planning to invite some friends out for lunch, trouble is, my mum wants me to invite a girl who was for a time (over a year ago) my best friend. Since then we've grown apart. It's not that she's a bad person, but she takes jokes far too far thinking that she's funny, when everyone else has stopped laughing long ago. She also can make highly inappropriate jokes, for example, she talks about how she wants to 'rape' my other friend, (something which I really don't think is a joking matter) she acts very creepily around her and grabs her shoulder whispering things in her ear, one time she went as far as to say 'I just want it touch _______'s genitals!??! Is that too much to ask?' She thinks she's being funny when everyone around her is feeling uncomfortable, today she said something to me that I felt wasn't appropriate and I didn't think was funny. I calmly asked her not too and she told me that she knew I was joking and to stop it. After one of my friends had to inform her that I wasn't joking, she apologised. But I want to invite her 'rape victim' more than I want to invite her. She always acts perfectly fine around my mother, who I haven't told about how she behaves when she isn't around because obviously it's not the kind of thing you really want to talk to your mother about when someone is saying things like that. My mum likes her and wants me to invite her. But I know that if she comes I won't be able to actually enjoy it. However I don't have a choice. No matter how many times people ask her to stop, or how many times she sees people looking uncomfortable, she still continues with it . I genuinely don't want her to come, but my Mum is making me invite her. I don't feel like I can tell my mum because it would be really uncomfortable. I also know that I can't just not invite her, or say tell that my mum that she's sick and can't come or anything like that because although she acts like this, I don't want to completely ruin our friendship, and I don't want to be a nasty person either. But if she does come, I know I won't enjoy my birthday. I am so stuck as to what to do, I don't think there is anything I actually can do other than just invite her and just accept that this is how she'll be. And I know that she will because I'm inviting 'rape victim' and I'm sure she'll do what she does before school, at break, at lunch, and after school. And I know that I won't enjoy my own birthday if she's there, but it doesn't really seem like I have any other choice. Please let me know if you have any suggestions.