Just stay away.
Your own child does not like this child.
You cannot change that Mom nor her attitude.
You either, give in and "make" your daughter be her friend. And this will then be a forced, friendship, and your daughter will get pressured and stressed by it personally and by other peers perhaps.
Or... you steer clear of this whole drama. And teach your daughter, that she has choices. And that, you are not going to FORCE her to be friends with this girl and Mom.
If you make your daughter be friends with that girl... you WILL also then have to, be friends with this woman and have her at your back, and you will be at her beck and call.
Just know, that all the other sane Moms... probably KNOW that this woman is off her rocker and that she will bad mouth anyone who does not do as she wants.
Do not be bullied.
And teach your daughter that too.
You either, become a "social worker" to this girl AND her Mom... and inherit all that toxicity into your and your daughter's life.... or you don't and you be appropriate for yourself and your daughter.
You cannot change, this woman nor her daughter.
The daughter and the Mom, needs to talk to the Teacher, and the Mom can get her daughter counseling or something via the school. If the girl has issues... which she does seem to have. It is for a Professional to handle.
I would steer clear.
You have to safeguard your daughter and her well being, and yours too. And you have to.... choose, what you bring into your home or not. Whether that be problems or not, from others that you know... are not good for your daughter or yourself.
You do not have to let this woman control and bully you, nor your daughter.