I think this is very difficult @CloverRiver
I suspect that the school with compromise because if his son leaves, they will lose money. Many private schools are closing because of the VAT impact.
However on the other hand, if you really are a blended family (you say you have adult children living at home) and you have a joint mortgage, surely you are in effect the same as a married couple and therefore, do you not want the best for your partner's son?
I can see you're wealthy- probably on 6 figure income as you mention the tax you pay - so why don't you want to help your partner with his son? His mum has disowned him , he has learning difficulties, he's had counselling- and he's just 13.
If you were married and this happened, what would you feel then?
Would you still refuse to pay?
I still say it's his father who needs to speak to the school rather than you asking on a forum, but part of me thinks you're lacking in compassion.
You also talk about 'if you split up'. [and you'd lose the money you'd paid out].
If you feel that's an issue maybe you shouldn't be with this man anyway?
Just throwing that out there as a question.