If I am understanding what your situation is correctly, when you moved out of state recently, she was required to leave all of her childhood friends behind to come live with you. Her problem may not be with the public school system at all, but rather with the situation that she is having to start over with making new friends. At her age, tween, that can be TRAUMATIC. Especially if she knew in advance she was leaving, she would have all this built up anxiety about what to expect and what she is leaving behind.
Girls that age can be particularly cliquish and difficult to be around. Especially if you are the "new" kid.
It also sounds like she knows that you are waffling on the situation, and so she is in a position to manipulate things to where she can just avoid dealing with a "new school". Frankly, I don't think it is more sinister than that.
She is being given (it sounds like anyway) a little too much control over her situation. And she isn't ready to have all the "privilege" (and REsponsibility) of doing that. It's just too much. Her solution, obviously, isn't to deal with it at all, but to avoid it all. "homeschool me" and I don't have to face the real world.
I moved in the 10th grade. I KNOW how devastating a move can be when you are leaving all the friends you have ever known behind, and have to make new ones. And the new ones already have established friendships, cliques, club members, etc. It is horribly stressful. You just need to put your foot down and help her through it. Don't let her decide not to deal.
You will need to make your home inviting for her to have girls over. Take her to some sort of classes or extracurricular stuff outside of school. Encourage her to sign up for things at school just to try them out. (band, chorus, a sport, yearbook, whatever). She will meet new people thru those avenues and have shared experiences with them... which, after all, are the basis for friendship.
Good luck.