J.N.
Wow, that's a loaded question. I grew up in a similar neighborhood and the girl across the street was my playmate from nursery school age. She was also this way, being sneaky and mean, but totally nice and polite in front of my mom. But she had all the cool toys (I guess mom & dad gave her toys in exchange for their time), so I always wanted to play with her. We'd get in fights, but after a couple of hours one of us started hanging out in front of the house hoping the other would come out and we'd eventually make up.
I think you're doing the right thing in allowing your girls to play with her supervised closely, even though the rest of the neighborhood won't associate. You're teaching your girls how to be polite and accepting of all people instead of teaching them to be judgemental like the other girl in your neighborhood who snubs your children based on association.
I think this is a VERY good teaching tool. Work with your girls to understand why people are so quick to judge and how inappropriate it is to judge people based on their associations. If the "problem" child were truly nothing but that, your girls would disassociate from her on their own. I think kids in general tend to know when enough is enough. Until then, you're keeping a close eye on them, giving that "problem" child some attention which is probably all she really ever needs, instilling in her a trust of adults and hopefully molding her as well as your daughter into upstanding citizens of the world.
Good luck.