Been there, done that. Based on my experience, your bigger problem will eventually be Mom X. We spent a lot of time coaching our DD and Girl X in how to find common ground, and DD began asserting herself more, by increments.
As DD started standing her ground and Girl X stopped getting her own way all the time, she started whining to Mom X, who decided to get personally involved by calling my wife and telling her how to be a proper parent. Not satisfied with this bad behavior, Mom X then decided to visit our neighbor (Mom Y) and try to sabotage DD's relationship with their kids by spouting a number of obvious lies, and enough bits of truth that Mom Y couldn't know from any other source that we knew when Mom Y shared the whole thing she was telling the truth. Finally, Mom X decided to try to work things out, but when confronted with her behavior with Mom Y, Mom X decided to try to lie and say Mom Y had made it all up. Strike three.
DD is no longer allowed to play at Girl X's house, and Girl X is no longer allowed in our house. If both girls happen to be at Mom Y's house at the same time, we don't argue, as we consider that neutral ground. What's important is that my child is not under the direct supervision of someone who would call her a brat, and my DW is not directly supervising a child whose mom will use that as an opportunity for offensive phone calls.
The X's has another child, and older boy. He has no friends. Girl X has lost other friendships due to her behavior, and like you, we were warned by other parents.
You've said your Mom X is judgemental, so she's probably already making (and may be reporting) negative judgements about you and your DD. You also said she's permissive, which makes sense, because kids who insist on getting their own way usually learn that behavior at home, where if they badger their parents enough, they appease. That's certainly the case with our X family.
