Hi whenskiesaregrey (This may be long as teaching and SEN are my passions)
I worked with pupils with special needs in a mainstream secondary school for two years and I know how you feel.
When it comes to stimming it seems to me you need to get your SENCO more involved; there is tons of evidence to show that things like tapping a pen or feet, rocking on a chair, chewing etc can help a child with ASD to concentrate. I've linked [ this video] so many times on here but it's so useful! (Proprioception disorder is very common in people on the spectrum). Teachers should not be stopping these behaviours!
My solution to your struggles (not keeping up with classwork, distracting classroom, annoying students) was to take direct ownership of the students' work/prgoress when it became extremely clear that the teachers were not going to. (But you'll need a brass neck and steel balls)
Can you hold your own detentions at lunch or put students in afterschool detention? We were allowed to and being TAs we were much more aware of bullying/teasing that happened on the "ground level" so it was stamped out very quickly.
I differentiated and adapted (one of my students was visually impaired) the work. None of my students could cope with the maths curriculum for their year so we did the year before or went over the basics again.
Even though technically it was "against the rules" I took my students out of many lessons to work elsewhere.
I set my students quite a bit of homework as there was very rarely class homework (I gave a certain amount of reading, spellings and maths with extra touch typing practice for my VI student).
I basically took over because it was obvious to all (including the students) that the bottom sets had been given the crappest teachers.
All of my students made years of progress (I was there 2 years and my 3 students' Reading ages went up by 3-4 years, Spelling ages by 2-3 years, my Irish Traveller students attendance went from 40% to 70% and 45% to 80%).
In conclusion you are 99% of the way there because you are clearly intelligent, motivated and you care. By taking a genuine interest in these students you habve already started to make a huge difference.