My 16 year old daughter has eBPD and doesn't want to see me

I’d really appreciate help and advice from anyone, please;

I am separated from my wife and our children, twin girls, have been for around 8 years
I’ve always tried to parent both kids exactly the same, as they are twins - may have been a mistake in retrospect?
We noticed at the end of last year that one of them had become addicted to ‘social media’ (Instagram, TikTok, Messenger, etc.,) and was on it around 14-15 hours per day
In January’20 I took them to see my brother/his family and we had a family chat over dinner - one of my daughters asked me what I thought she should do with her life? I replied that it wasn’t for me to tell her but that I would try to support her in whatever it was that she decided - she didn’t like this
I returned them to their mum after the weekend away and haven’t spoken, texted or engaged with her in any way shape of form since - she’s blocked me on all social media and ignores all my texts, calls, FaceTime, etc., and stays in her room when I collect the other one for weekend visits
Me/my ex agreed to impose restrictions on the use of her phone to stop her staying up until 4am/5am each morning and going to school exhausted but I suspect in retrospect this wasn’t done in a good way and my daughters probably sees me as the bad parent here (I pay for their phones on my phone account)
She was referred to CAHMS shortly afterwards by her school and after determining her to be competent asked her if she wanted me to be included in her treatment, to which she replied ‘no’
My relationship with my other daughter has continued, pleased to say that this is quite positive
I had a call from my ex a week ago to tell me that the daughter who won’t see me had left her house to take an overdose in the early hours but luckily had called 999 and had ended up in hospital (I was also told at the same time that she’d tried this at home a couple of months earlier and that they’d been aware she’d been self-harming for a few months, although my ex had asked my other daughter not to tell or discuss this with me, which must have been very hard for her)
This led to a consultant at the hospital diagnosing her with (emerging) Borderline Personality Disorder and if it were possible to take a positive from this, this has started to help me to understand that in fact one of my daughters is suffering from a dreadful mental illness - but I am so sad that I can’t do anything to help her; I know its not about me but its hard to just ‘sit back and wait’
She has many of the characteristics of the illness, moody, explosive tempers and has had a few obstacles to overcome - starting with the separation of me/my ex wife, open-heart surgery (for that daughter), changing house, school and friends, the loss of both my parents and now this …
But I can’t understand why she won’t engage with me
She seems to use TikTok as a way of expressing herself and although she has agreed to have her mum involved in her treatment they still have huge rows, some of the text exchanges (she basically doesn’t emerge from her bedroom so far as I can tell, except too go to school) they’ve had are awful
When I got the call last week, the first thing a parent wants to do is go visit her in hospital, give her a hug, be a protective father - I was asked not to, because she continues to say that she doesn’t want to see me
The consultant at the hospital called me and we discussed her issues and I am now in a position where I can’t do anything to assist her, can’t see her, all it has been suggested is that I write to her (letters, not texts) that ask if we can sort the problem and that I’ll always be there to support her in whatever she does and whenever she might decide to see me again

Does anyone have experience of this? any advice? any suggestions as to what else I can do, please?
I miss her so much and am so grateful that the relationship with me other daughter is so good …

And finally, it has helped me to understand things a little bit more and as I have a history of fund-raising for charity I now have a new purpose/target and will be looking to do some fund-raising once all this covid stuff gets under control and we are able to get back to something closer to what life used to be like

My heart goes out to you. I have a thirteen year old with very similar issues and my experiences sound very similar to yours apart from mine is on the same household as me and I am the mother not the father. My daughter often does not want to know me and shuts me out. Also the explosive tempers and the tik tok. She took an overdose aged 12 and was in hospital after it. I have no words of wisdom but I do treasure the moments when she is my little girl again. How long is it currently since you have been able to talk to her ? I find the shutting out comes in phases with very small times when she is reasonable again. I try to focus on the positive and avoid being critical when she does things that I don’t approve of. Hope it helps to hear from another parent at least…

I have a friend whose daughter told the most awful lies, as if she just didn’t understand the concept of “telling the truth”. Then her dad died, she fell out with everyone, and finally she was diagnosed with BPD, in her early 20’s. Now stable on medication, she has a partner and baby.
You can’t help someone until they want to be helped, and having no option but to sit back while your child is pushing the self destruct buttons must be so hard.
However, you have done the right thing, enabled her to get the help and support she desperately needed, rather than bury your head in the sand and ignore what was going on.
I believe that parents always have a joint duty to do the best for their children, whether or not the parents are together, whatever age their children are, as adults as much as when they are children.
If you bring a child into the world, they are your responsibility as long as you live.
So I don’t think much of your ex keeping important things from you!