I've never been in a situation like this, so take my advice with a large grain of salt.

I'd definitely take the high road. Interact with the daughter in front of her mother to prove that you aren't this horrible monster she's worried about. If she seems that her daughter enjoys spending time with you and you are a positive influence, she might begin to change her ways. Just always be polite and friendly. Heck, maybe even tell your boyfriend's mom how much you enjoy spending time with the daughter, and maybe at some point in the future the two of you could go on a shopping trip or something to bond. But be sincere.

I also think your boyfriend and his ex need to work out a new visitation schedule. It's not fair to him to not get to spend time with his daughter, but I also understand she needs time to spends with her friends. Is there any way your boyfriend could host a sleep over for his daughter? Or take her and her friends to the movies and pizza? I really believe they need to figure out a schedule that works for all three of them... Not just one or two of them...

Good luck!