I think you have answered your own question in this matter. Firstly, what she did is very wrong and if by not talking to her at all, you're doing a good thing. Secondly, you said that it would be selfish of you to not invite her. But the thing is it's your birthday party as well, and I feel you reserve the right to invite anybody you want. If your other friend doesn't really mind, then I say you don't invite this girl. Fact of the matter is: it's your party as well, and the rest of the year should not care who is there, and just be happy with the fact that you arranged something for them.