P.G.
I agree with the other moms - I like the idea of having a gathering of the other girls for special time. Definitely use it as a teaching moment for your daughter, and the other girls if they'll allow it.
The girl is acting as expected considering the behavior of her mother. If her mom said invite X number of guests, mom should either have allowed a few more kids or not let her daughter dis-invite people. She's the one with the problem, but unfortunately you have to deal with it :(
Good luck, and have fun with the other girls, maybe even the moms - who knows, everyone might end up with some nice friendships after all this.