Viking, its brilliant DS was able to access inpatient care so close to home. In the U.K. children and young people often face being admitted miles away from home. It really hampers progress. Your system sounds just as stretched directing parents to look for private therapy.
Zoo, I'm glad you managed to speak to a CAMHS psychiatrist, and you know how move forward. Monitoring weight/height with the school nurse is a good idea. Do you think it's verging on an eating disorder?
Katalex, If DD talks of suicidal ideation again and you think she is serious you need to take her to A&E. I do partly agree with the GP, there is a difference between feeling anxious and anxiety in the medical sense. Though you could ask again for GP to refer to CAMHS. Long term most LA's want consultant evidence of absence, or at least a CAMHS referral sent.
Yes, ask to meet with SENCO. Schools do have limited access to EP, but to be blunt, it is very, very unlikely they will use their limited hours on your DD. I'm not saying it should be that way, but that is reality. The only way to force an EP assessment would be to apply and be accepted for an EHCNA. Something you should consider applying for if DD really can not manage school lessons. Though I do think student support is preferable to being at home. She is then there, if she isn't in school there is limited support school can offer. And the longer she is absent the harder it will be to go. But when there they could offer a mentor/key worker, do they have a counsellor, do they offer ELSA, social skills or SALT interventions, time out card, early lunch pass, ability to move between lessons 5 mins early, laptop, an IEP, help with organisation, use of 5 point scale, ear defenders/noise cancelling headphones, looking at placement within the classroom, assessing for GCSE access arrangements (though the deadline for those that need applying for is February and needs to be normal way of working so pushing it), pre teaching some subject specific content... the list really depends on why DD is finding certain lessons too much... And they're boring and she doesn't like the teacher/content aren't really acceptable.
Re ASD assessment, make sure whoever you go to privately is a) experienced in diagnosing high functioning females and b) preferably also works in the NHS. The later is because sometimes LAs/schools will question private diagnoses. The Lorna Wing specialise in high functioning women, though they are £££.
Poor executive function and cognitive processing can be part of ASD. Normally this would be assessed by an EP. Depending on how ASD assessments are undertaken in your area it may include an EP assessment, or it may not.
Writing down how she feels for student support is good.