I have two autistic DC and as I was reading your post-OP, I was thinking your DC is autistic - as everyone else has said, it's classic masking behaviour and you are their safe space. It's scant consolation right now I'm sure, but it means they feel safe with you and that allows all the stress and corked-up pressure to come fizzing out.
In our area you also need the backing from the school. Our paediatrics dept has a checklist and if any referrals are made without the items on the checklist, the referral is thrown out. It's a really terrible system, particularly when some schools are very poor at identifying autism (especially in girls, but that's another story).
One of the things that you need in our area is a report from an educational psychologist. The school can arrange this assessment but it costs them £££ so they won't do it unless there's a clear and obvious need. There's usually a huge amount of pressure on Educational Psychologist (EP) services so they might not be willing.
However, it is very possible to get a private assessment done by an EP and I think this would be an excellent starting point. A good EP report can be absolutely invaluable and carries a lot of weight.
You need to start building your evidence too. I know this feels horrible and you shouldn't have to do this, but others need to see what you do. Start keeping a diary of behaviours and activities - you might start to spot a trigger that you hadn't noticed before. For example, worse behaviour on PE days (can be stressful activity for some SEN children), or on Mondays (transition after the weekend), or on Fridays (masking is exhausting and gets harder as the week goes on). The other thing a diary will do is establish a continued pattern of behaviour so you can show it's not just a flare-up, or a temporary reaction.
Video/audio evidence will also be very persuasive. Doctors and schools won't usually hold this type of evidence on their files, but you can show it to them. The EP will find it useful to see what your DC is like at home when they're not being observed. Our EP (not private) came to the house and spent three hours there. My DS (he was 5 at the time) sat in the garden the entire time and refused to enter the house while she was there (she was lovely) even though he'd seen her and engaged with her at school.
Our EP report for DD outright suggested that a diagnosis of autism may apply, and that she needed urgent assessment by paediatrics. Do your research and find one who specialises in autism.
I'm sorry it's so hard to get your DC's difficulties taken seriously. We changed schools as the first school were similar re school refusal. I used to physically carry DS kicking and screaming at the top of his voice and the SENCO said to me "You really do need to get tougher, Mummy"...... But other schools are amazing and completely understand the very complex ways a child with SEN can present. Are you sure they are at the right school to meet/support/understand their needs? You're describing a very classic autism presentation - and the school's response is to threaten SS report?
Anyway - the key thing is gather your evidence, and get professionals on board privately. The EP is a really good place to start. I would also suggest a speech and language assessment privately. My DD didn't have obvious language difficulties but when she was formally assessed, they identified lots of issues that aren't necessarily obvious to a parent. Good luck.
PS - And no, you're definitely not the problem!!