I think, in the short-term, the issue of the chairs is probably the most 'winnable'. You have a specific sensory problem with one type of chair. The sensory problem is due to your autism, so it is disability related. You do not have the problem with other chairs, that are already available in the school. I think that a comparison between the relative unimportance of marking or scuffing a floor and the wellbeing of a disabled member of staff is quite obvious. I would make your reaction to the sound of the chairs comprehensible to the SLT by comparing it to something that upsets most people, like knives squealing on plates, or fingernails down a blackboard. Saying that you experience that type of unendurable sound multiple times a day may lead to some realisation of the effect it is having on you. Employers are legally obliged to make reasonable accommodations. If the school managed to swap chairs in one direction, they can hardly argue that moving them in the opposite direction is not 'reasonable'.
Was there any mention of 'discrimination on the grounds of disability', in your HR meeting? Playing that card can often get results. No organisation wants to be seen in that light, very bad for their reputations.
Are you a member of one of the teaching trades unions? If not, I would be inclined to join. Most branches will have a disability officer and you can ask for a 'case officer' from the union to attend any meetings you have with your management.
I worked in universities and my union (UCU) was very helpful when I had a dispute with the university over my contract. That my case worker was a lecturer in employment law helped!
