I was a disabled pupil in a private school, it was the best thing that ever happened to me.
A smaller calmer environment than the state schools could provide (even then 30 years ago, schools are much bigger now), I escaped from the bullying and 'feeling different' that I had experienced (aged 13 when I moved), no one in my school was snobby, and no one knew who paid fees and who was there on a grant. My best friend, who is now a very well off partner in a large legal practice, having got a 1st at Oxford, was a working class girl on 100% grant, but none of us knew it then.
You will, I firmly believe, get a better education for your son in a decent private school, who will be more able and willing to accommodate his needs. My school once re-arranged a whole terms timetabling for the whole school to ensure that I could come back to school in a wheelchair and do all my lessons on one floor after having surgery. (ok, now the school will probably be completely accessible, but as I say, this was 30 years ago.
If you and your ds would like to try for the scholarship, then go for it.
and as for "people in my family (and DH's) don;t go to private schools", well, that's rot. Once upon a time people from working class families didn't go to university, holiday abroad, go into restaurants.......
It's no good looking for a 'classless society' if people say "people in our family don't do that"
G F I, and very good luck.