Have you looked at Egerton Rothesay in Berkhamsted? The website is poor but I would encourage you to find out more I.e. don’t be put off!!!
St Christopher School in Letchworth might also be worth a look. I know a child with HFA doing very well there. Ironically, what drew them there in the first place was not having to wear a uniform (child has massive sensory issues).
The problem you will face is that private schools can pick and choose who they take on, unlike state schools. It’s not right, but ask yourself why should they take on your son when he can’t perform in assessments when plenty of other children can. They are businesses (arguably charities…..) and are rated on results. It can just be too “risky” to take on children with SEN.
Private schools pride themselves on having a huge array of extracurricular activities that children can join in. You’ll see them playing sports, all sorts of music, drama, art, trips (including residential) etc. but you say your son is unwilling to participate.
There’s also the subject of money. Who pays for the extra support your child needs navigating break times etc and where does that staff member come from?
So, your sons needs may well be too far away from what many private schools can actually offer him. That’s why I would encourage you to think a little more widely and have a look at the two I suggested. ER in particular would offer him everything he needs. You’ll see fees are very expensive but a lot of children there have their places funded by the LEA when it is named on the EHCP.
I do feel your pain though. A family member has SEN and finding the right place for them has felt like a battle that had taken years to get them in the right place for them.