Several thoughts on this situation.
I do not feel that switching school is bailing your child out.
I do not think a person needs to experience a situation that is bad for a long time to recognize that it is bad.
If your child spend a fulls semester in a school before switching, they will then be forced to adjust to a new school as the new kid. Switching now will allow her to fit right in with all the kids.
16 is a young 11th grader. I have to assume your child is bright, able to navigate her life, and could use a little support if she feels a situation is bad. Do you NEED her to tell you what the problem is before you support her?
Those that claim that life is full of people we don't get along with and should just accept it should possibly think of things a bit different. If WE didn't accept those that were not easy to get along with, maybe they would be forced by peer pressure to CONFORM to being a better person that is more easy to get along.
What age do you THINK a child should have the right to determine their own fate? We are supposed to be raising self confident soon to be adults. A 16 year old, 11th grader, should have the right to make some choices on their own life.
If you were paying for the school (such as college) and your child said that they could not learn in this environment, would you continue to pay the college so you were not bailing her out and force her to learn to get along? Put a price tag on this situation and see if it is still worth it for you to continue to fight over it.
My children are in a digital academy and I love it. They are free to move through school at a pace that they set for themselves, and each of them are 2 full grade levels above their age. They are not embarrassed by other children to feel foolish for learning or forced to think of what their clothing cost is compared to their studies.
Last thought, teen suicide is VERY HIGH. While I don't think we need to give in to every demand a child wants, I do think that we need to support our children with decisions on they should be able to make for their lives. I do not think we should force them to ENDURE until they no longer feel they can endure and decide to END it instead.
Hope that you and your child are able to work this out, and that it makes the relationship you have together stronger.