I recognize that with a private school they may say, "You are free to leave." But the flip side is, YOU pay THEM to educate your child and not bully her. If the principal and this woman are in alliance, then maybe it's time for you and other parents to go to the next step and go above the principal.
Further, I would be very reluctant to put my child in that teacher's class next year. Can you imagine? Forget academics if she's constantly chipping at your daughter's self-esteem. Your daughter is already quiet and reserved. I'm not saying that every day needs to be sunshine and roses, but I used to work for someone like that and if I as an adult felt depressed, what about a child?
If this cannot be resolved, then I would push back on my husband. Academics are important, but at what cost? Is there truly no other school for this child? Your daughter already has a full school day, music lessons AND now required tutoring. That's a full load for a kid, nevermind having to deal with someone who comes out of her class to target her - which is just what you witnessed. What does she do when people aren't watching?
Your mama sense is tingling because something IS off here. It sounds like this woman has the principal's ear and is using her influence to get what she wants. I'd be concerned.
Edit: You say your husband says you are staying "no matter what" so that gives them no incentive to change. You say you cringe at the thought of your child spending 9 hours a day with that person, but your husband doesn't want to consider an alternative. Your child is already shy and timid and anxious about being perfect. Are they looking for the best a child can do (sometimes less than an A) or are they looking for high scores so they can tell prospective parents how "good" they are? An entire school of As would make me question the grading, personally. Another parent is already warning you not to go anywhere - I would take this information under serious consideration. Not that I think kids should travel all the time, but what if she misses for a legit reason? Is this teacher going to inflict her wrath? I think there's more going on at this school (and in your home) than just this one teacher or incident. Do they really want to work with you or do they just not want to lose your money? Does your DH want a solution or a bandaid?