My DD, then 9, went to a spend the night party once where the invitation asked that the guest bring $5 for snacks because they were going to the movies (drive in). Since it was a small amount and it was a spend the night sitution, I didn't mind it too much. However, any more than that will be coming out of the gift budget. I'm pretty much with the camp that if you can't afford to cover your guests cost for a party then you need to scale it down. Rather tacky otherwise.
OP, I would let my daughter go if she was willing to pay for it herself since she wants to go so badly but reiterate to her the reasons why you are not willing to do so. When the money comes out of her pocket maybe it will be a learning experience for her.
See now I think that is tacky. If they couldn't afford to buy snacks then I think they should've made popcorn and brought juice boxes or skip the movie. I'm just firmly in the camp of if I invite you then I pay and vice versa. I've never encountered an invite to a party liek that though, I would have probably just handed over the cash b/c I'd be in shock, lol.