Hi Cat,
I was a lot like your daughter at that age (30 years ago!!). I'm a Highly Sensitive Person (HSP) and an introvert. I score highly on the 'sensory overload' elements of Aspergers tests, but very low on the other elements, as I interpret lots of information from other people, so I'm very good at reading other people/social behaviour, but find it exhausting being around people because I can't switch off from all that sensory input.
I can highly recommend Elaine Aron's books, and would also suggest 'Quiet' by Susan Cain (there's a lot of overlap between HSP and introversion).
I completely agree that you need to keep pushing for CAMHS if she's self-harming, but in the meantime, some things that have helped me over the years (and I apologise if this is all stuff you've tried already):
Walking: Children need 60 minutes exercise a day, but lots of that gets done in busy, noisy playgrounds, or in classes where there is structure and the stress of trying to do it correctly and please the instructor. It helps to get out walking, preferably in nature, with someone who isn't going to expect a lot of interaction. As a child, I spent time walking in the forest with my mum every weekend, and walked home from school every night from year 3, and this was very soothing and calming, and allowed me to make sense of my own thoughts (I think better when I'm moving!). Still now, if I'm feeling anxious and stressed I'll get out for a walk.
Also, it's very intense if someone sits you down and asks if something is wrong/ tries to get you to talk about feelings - it's almost as though the feelings are going to be worse if spoken out loud. I would struggle with that even now. It's far easier to talk when walking side by side with someone, and when given time to order my thoughts before speaking and raise things in my own time. This has become a family tradition, and even now with my own husband and son, if one of us is struggling or there's an issue to discuss, it'll get sorted when we're out for a walk.
Food: Lots of children this age don't get enough protein in their diet and have a very carbohydrate heavy diet (because that's what they enjoy eating - bread and fruit, essentially). This excites the nervous system and causes big swings in insulin/ blood sugar. Some neurologists are now even recommending ketogenic diets for 'excitatory' conditions like epilepsy and anxiety. This is obviously unsuitable for a growing 7 year old, but ensuring adequate protein and fat intake, and minimising simple carbs, might help calm her sensitive nervous system, and prevent the compounding problem of swings in blood sugar.
Meditation: The Headspace app has a kids section - we use this a lot at home and have a few minutes meditation after school and at bedtime (literally 2-3 minutes for my 7 year old). This is a great way to teach children with a lot going on their heads to be able to get a break from all the thoughts and feelings.
Time alone: Who else is at home? Does she get to have peace and quiet, playing alone or reading in her own room after school? If she has less sensitive siblings who always want to play, this can be exhausting and overwhelming (I speak from experience!!)
Noise/ light at home: Lots of people who have visited us comment that it is really quiet on a morning in our house. We have no TV or radio/ music playing before lunchtime at least. My parents always had the radio on in the mornings and I remember finding it really irritating, and then setting off for school already feeling 'jangled'.
I think year 3 can be a tricky time, because kids start picking out people who are 'different' and recognising their own 'differences' from others. There's also the start of the urge to fit in and be seen as 'cool'. The playground banter becomes more adversarial and can tip into bullying, especially if a child finds it difficult to come up with good verbal 'comebacks' in the moment (another introvert trait). I started being much more anxious and dreading school around this age, despite performing well when I was there.
If it helps, I also was great at school, could hold it together, but then struggled and felt bad that I couldn't control my feelings anymore when I got home. I longed to be one of those easygoing, sociable kids that just didn't seem to get bothered by stuff. As an adult, I've learned to use my sensitivity for good. I hold down a challenging professional medical job, have a loving and supportive marriage and family, hobbies and a few good friends. I wouldn't want to be any different to how I am. However, I need to prioritise good self care (all the stuff above), and very quickly get anxiety symptoms again if I don't pay attention to these.
This is a very long post! I hope some of it helps. Thank you for recognising and acting on your daughter's distress, you're a great mum to be taking it seriously and not telling her off for 'taking things too seriously' or 'thinking too much'.