I think it's time to talk to the parent. It's their job to educate their child about manners and how to act when they're with other people.
They need to teach their girls that they are separate people and won't always get to do things together.
If the parents do their job then this sort of thing won't happen as much. They will simply let one do things like sleepovers and the other stays home because they can't mind as well and aren't as well liked.
If the parents don't want to let them do everything together then they should say no, that both girls go or neither one. Then it's on them and not on you.
So don't let your girl do the invitations for a while. If you ask the parent then they have the option to say yes to one or no. Then you have the opportunity to educate them on how their other daughter acts in your home and that she's too much trouble.