I work in a school and I would recommend you contact the Local Education Officer (LEO) to discuss issues of the failure in the duty of care to your child. It's not strictly the procedure (you're meant to complain to the Head, then if not satisfied, the Chair of Governors, then if not satisfied, the LEO) but these are serious Health & Safety issues here and it seems like the Head is not taking responsibility. They will probably want to see you with the Head. The fact they are involved shows how seriously you are taking this and I don't think the Head sounds like she has up to now. The Boxall profile is all very well, but health & safety must be addressed first. This is basic common sense, it must be so frustrating for you.
I agree that probably your relationship with the school has deteriorated to the extent moving schools would be best particularly as these problems seem to be denied and ongoing. At Key Stage 1 (Reception, Year 1, year 2) the maximum number in any class is 30 legally, so that's hard to get round and probably on used in EXCEPTIONAL medical circumstances. Older than that and the Head can go up to whatever they think the classroom and teacher could take. It's worth putting your name on waiting lists as people move house all year round so you never know. Mid year joiners tend to ring us direct and we sort it out - there's no need to go the Local Authority unless it's applying at an age children would normally start at the school.
Good luck!