Sorry Dadu, I had to jump ahead after reading the first couple of pages.
So pardon me if any of this repeats others' thoughts and suggestions.
First - on "Caring" It seems to be a theme you kept coming back to, she doesn't perform because she doesn't care. So, have you asked her what she does care about? Is there a way to fit that into her class life? Maybe she doesn't care about money so the match sections about adding money don't connect with her, if she likes turles could she add turtles instead? see what I mean, seems like this child needs a creative approach and I don't have the answer for HOW to do that, just that it might be worth a try if you can find someone willing to take that approach with some experience doing so.
As for Christian school - just a couple of thoughts - first if it's something you are interested in do not write it off solely because of financial concerns. Many, if not all, have financial aid available, I'm on a foundation for one, and it's not only a factor it's part of our schools mission to reach out and care for those who want that eucational experience but cannot afford it. For us, even with full tuition it's not that expensive, but that varies a lot from place to place and school to school. So, if you're interested, persue the interest and ask questions, express your concerns about money, be frank about your situation, and you might find help. Now, the second thing is that some private schools don't have the resources to handle kids who need special attention very well. It's simply a resource issue. So, if you think the nurturing environment will make a difference, then it may be worth looking into, but you may not have access to as many resources. I hate to generalize.
It seems like a course of action would be to meet with some of the schools you are interested in and see what their ideas are for your situation. Maybe one of them would have an interesting approach, and it would be easier to repeat a grade in a new school than the same one.