Is there any way you can prove the teacher is being discriminatory? Is there a different class in the school with a more understanding teacher? It sounds to me like the teacher is impatient and not very understanding.

Also, it seems like the main problem is her not wanting to handwrite. Does she know how to type on a computer? Is there any way she could use a laptop for notes instead of writing by hand? I learned to type when I was 8, I loved it.

I remember note-taking being a nightmare in school. There were too many notes to take in too little time and I could never write neatly even if I took my time writing, so when I tried to keep up with the teacher's notes I couldn't read anything later, eventually I just started pretending to take notes but was really doodling because I knew there was no point. I could either take the time to make the writing legible and miss some of the notes because of being too slow, or I could try to write them as fast as possible and not be able to read them later rendering the notes useless in either situation. Lucky I was smart enough to pass without studying, but I wish I'd had some help to be able to get the grades I could have.

Unless you can find a new class in the same school with a different teacher or find a way to prove that the teacher is actually the one who needs to be assessed, not your child, then it may be best in the end to find a new school. If she has to go to a new school she still may be able to keep in touch with some of the new friends she has made.

Also, if you can convince the school to give her another chance, talking to your daughter about the fact that she may have to leave the school she likes unless a solution is found, might help her to realize that she can control her life more if she can control herself, sometimes incentive is more useful than anything else. Also might be a good time to talk about the behaviors that the teacher has issue with and brainstorm some possible solutions.

Try to find out if it's even legal for them to kick her out like that, it doesn't sound like she's done anything to justify expulsion and it sounds like a failure on their part that they can't provide the support she needs. Maybe the teacher needs to have some specialized training, like on empathy, and special needs kids.
Good luck.