As soon as my d d has a friend that girl will get invited to her house!
This will be the mother's doing, no doubt. Is it this girl's mum who you are friendly with? What's she like? Tell us a bit more about your friendship with this mum. How does she treat you? Do you have things (money / nice husband / nice house / nice car / better education) than them? Is you dd more talented / better at school etc than the bully girl? This is all very likely due to the mother's competitiveness and feelings of insecurity.
I know a mum just like this. She has to be better than me and her kids have to do better at everything, have more friends etc.
This mum is unhealthily competitive, and derives her satisfaction from her kids outperforming other dc.
Whatever activity my dc start and are seen to do well, she will instantly get her kids to start the same thing, think piano / ballet /language lesson absolute whatever. If my dc has a playdate with a new friend, sure enough this crazy mum will arrange one the next week when she or her dc had never been interested before. Whenever I speak to another mum at school, she will barge in, disrupt and focus the interaction on herself.
Pathological levels of envy, I think. .
Funnily her dd is starting to become just like her
.
Best thing to do is play her at this game but also speak to school. Exclude her dd and teach your dd to put this girl in her place.