OP, you poor thing. This sounds quite typical of our public services atm.
First thing - reassure yourself and your daughter that this is about resources and resources alone. It is nothing to do with her on a personal level. They have hardly any resources or people available so the service is just not there for many people. They send people away all the time and it is not because that person is not suffering enough.
Do you have any savings or access to funds for private support. That CAMHS person was not the right person to help or advise. They sound pretty incompetent.
You suspect your child is ND? If you have the tiniest sense of this, you will be correct. Mums always always know, even subconsciously. I think you believe she is. Just getting an answer to this can be very profound for a confused, distressed teen.
Can you afford a full assessment for Autism and ADHD? You can search online and look for accredited professionals who follow the NICE approved guidelines. There's something called Right to Choose I think. I hope other posters can chip in here with advice how to access that. It's where you appeal to the NHS to assess. You need to be assertive though. You insist you believe she's ND. Many young women are given all sorts of incorrect diagnoses when they're actually Autistic/ADHD.
Online CBT sounds like a pile of shit but you may not have any other options I appreciate.
Would she be willing to consider any medication such as SSRI? These are antidepressants and the GP prescribes. I am not entirely clear if this is something they'll do with under 18s. Young Minds can give advice. I called them alot when my ND son was saying concerning things. If these were prescribed, you'd ask for the lowest dose possible and I personally would use a pill cutter from Amazon and start with half the dose to reduce any side effects. I'd ask Young Minds for advice on all this. I don't know the data on under 18s and SSRIs.
Is there any way to fund private counselling with an ND aware female therapist?
I'm so sorry that everything involves paying. I don't believe our public services are fit for purpose tbh.
Every ounce of invaluable support costs money.
Young Minds were just amazing when I called. I will go look for a link for you.