L.P.
The way I would handle it is to organize play dates so to speak. I would tell my daughters to invite just the ones they wish to play with over for a snack after school. Make a batch of cup cakes and buy some things to decorate them and let them decorate. get some bananas and markers and then buy a cheap pak of misc decorations and have a banana decorating thing. buy water balloons and fill them for them to have a water balloons fight with. Just small cheap things for them to do. By not inviting the bully and inviting the others they will come because you can give them something special to do and let the others know if they ask and only if they ask that when she (the bully) can start playing nice she is welcome to come but that her behavior is not allowed in your home. If the mother contacts you let her know the same. She will either quit so she can join in or be lonely and will come around on her own. But this way the other children are being invited not just going outside and saying hey lets go to my house. Just make it a couple times a week. If you are able and up to it have a sleep over with the others. I have found that bed rolls or sleeping bags, and some movies works great. Pop corn, with grape juice mixed with sprite to drink and a box of chocolate chips cookies from walmart bakery they will entertain themselves. Trust me by not inviting the bully her mother will take it personal as if she was not being invited to functions for adults and will address what the problem is. It may seem cruel but it works had the same problem with our son in the 5th grade. We had a party at the house and invited everyone from his class except him and his mom called to say he was very upset and crying and wanted to know if it was an oversight. She started out very rude telling me how wrong it was and then I informed her that I would not invite a child to my house that constantly name called and pushed other children around cause I wanted them all to relax and have fun and if he can learn to get along and be nice I would be glad to include him. He stoped immediately and the boys actually became friends. Good Luck.