Welcome to the Forum where you will see many have had to deal with this very worrying problem.
I guess in some ways you are like the captain of a ship who can see what is needed to get to the destination, but has to deal with a suddenly sprung a leak. Priorities change drastically.
I can quite understand the need to prepare your daughter for later life, it is something every thinking and caring parent tries for.
I would imagine that lack of self-esteem, self-harm and your daughter's other problems are made worse by pressure, and this is why your medical advice is to soft-pedal studies or the school environment. The immediate need here may well outweigh future scholastic requirements.
If the psychologist has recommended a psychiatrist I'd encourage your daughter to go. The fact you do not see much evidence of proper coping strategies at home is not surprising. They are not easy to accomplish and harmful ones have to be discarded, something even harder.
I can only indicate the approach I'd try to take, which is no quick fix. I'd try to get my daughter to see me as someone she could talk to without argument, someone receptive and if necessary non-judgmental. Comfortable.
At the same time try to be a companion, entering in to her activities and doing things together - things she has enjoyed in the past or wants to do. Easy companionship and hopefully in time a measure of willing cooperation. Anything that builds up her self-image such as asking for and maybe deferring where possible to her views may assist.
When all else fails just let her know you will always be there and leave it at that.
Yes from your point of view it is 'going easy' on her and you could see it as bad for the long term, however if your daughter is in such a fragile state she self-harms then that may be the only realistic option at the moment.
If you could get her to talk to someone when she feels the need to self-harm that would be great. The trick I suppose is her having enough confidence in her reception to feel able to do so. There are a fair number of other techniques to minimize this behavior, hopefully something she can talk about with her medical team.
The fact she has a loving concerned and thoughtful parent and the resources of a doctor and psychologist is a really great basis for improvement, and most anxiety related problems do respond well to the correct treatment in time.
I do hope you will return and talk more