Op - firstly you will need to let the 'teen drama/CAHMS pack' go - there is a tendency on MN for posters to collectively focus on an early in thread perception and then gang up on the OP. No matter how much you protest - they will latch on and not let go. You will just end up feeling worse than you already do by engaging with them.
I totally understand and am in a similar situation. Our eldest DS is now a bit older than your DD but similar age gap with 2 younger siblings.
My DS had by all accounts a normal upbringing, verging on indulged. I had PND after he was born, it took me 6 months to adjust to being a mum and after that I totally over compensated and inadvertently gave my child resilience issues by always being there with the solution. From about age 13, he started to become more and more withdrawn and difficult, but we didn't give up. We carried on being there, morphing, changing, adapting to his needs.
Fast forward to 1st year uni, it all fell apart. Self harm, 2 attempts to really hurt himself. Lots of self loathing and thoughts of inadequacy rooted in what he perceived to be a terrible childhood. Like you I was there EVERY step of the way to support, love, cajole, listen. I love that kid with all my heart and he was slowly breaking me and punishing me for things beyond my control (sibling rivalry). At this point he was in a toxic relationship, and in a toxic friend group - I think they fuelled each other in lots of not nice ways.
One day (he was about 21 at the time), it dawned on me that I was feeding his anxiety and making everything worse by being hyper-attemtive. What's more, I was in serious danger of alienating his siblings, who quite rightly had lost patience. I didn't do anything other than make a mental separation - he likely would not have noticed. I made my peace with my truth - his childhood was not toxic and vowed to be there for him in a different way.
He is 23 now - I don't think he will ever have a relationship with his siblings - it's just too far gone. Luckily, after lots of intervention, he seems to have turned a corner and is no longer angry with us. In his own words, he was furious with his life - he didn't know why, he just was and wanted to blame something. That was us.
We (DS, DH & I) do not know where this boiling, festering anger came from. We can hypothesise, but he really can't pinpoint it. He is certain though, that a collective identity was fostered at uni and they all focused on terrible childhoods.
It hasn't been easy. There were many times I felt like l had been pushed to the edge. My advice is that you need to make a mental separation. Let your DD know that you love her, but NOT at the expense of her siblings. You can't control her anger and need to lash out - trying only makes you more of a target.
🌺🌺🌺