I am really reluctant to join this thread, as I understand you are at your wits end, but I don't think your experience is unusual and I think you have quite high expectations of what schools should offer.
Your DD has had counselling in the past, you are paying for Psycologist counselling now, and she is on a waiting list for more school counselling. IME school counselling is just that, someone to talk to, and there for emergencies. I have never heard of a school having a CAHMS counsellor. The private counselling she is having is perfect for her needs.
I agree, the school seems very unbending with regards to dropping the Alevel. There is absolutely no need for 4 ALevels, not even for Oxbridge, although helpful for Medics and VetMeds, otherwise 3 good ALevels are perfectly sufficient. Is she doing an EPQ as well??
I think your GP was very ill advised to "gossip" about a particular school. The sad fact is that ALL schools are experiencing increased mental health issues, anxiety and self harm. It is a societal problem, and certainly not limited to either girls schools or private schools.
Finally, I wish your DD luck going forwards. She is obviously a very bright girl, and your are supporting her well, but I would gently suggest that there is only so much a school can do, and at this point, it may be better just to let things calm down, see how the three ALevels go, when she does her exams this year, finish the counselling, have a nice summer break, and see how you all feel going forwards.