Kirky, I can't offer direct advice, but just to give you an example of how a 'bad school fit' can affect a child...
DS, at 5, in Year 1 in the first school he went to, had become a selective mute. The SENCo was involved, and brought in an Ed Psych. He also had many 'ASD'-type behaviours - rocking, failure to make social contact with other children etc. Like your DS, he was very academically bright, but the muteness and othger social issues were getting in the way of his learning.
Following a house move and a short period of Home Ed (because he was going downwards fast, and I felt that gave me a good means of stabilising the situation before he went back into school again), DS started at a new school.
Both were state schools, both Ofsted 'Good'. One was small and in a village, the other large and in a prosperous town. The contrast in DS when moving to the new school was extraordinary. He emerged from his first day talking 19 to the dozen (his friends find it absolutely hilarious that he was ever mute, as they have never seen him so), the ASD traits lessened (he is still a child who needs absolutely explicit teaching about e.g. emotions, social norms etc as he does not 'read' them at all, but the tics and the rocking and the hiding under tables all stopped) and he rapidly developed a circle of close friends.
The second school is 'child focused' - it looks at who the child is, and what they need as an individual, first and foremost - not to 'mould ' them to be some uniform 'perfect child', but to be the best that they can possibly be.
So a change of schools could work wonders, and worries about SEN (DS has never been on the SEN register at his new school as they waited to observe how his difficulties showed themselves in his new setting ...and they never did..) may evaporate. If he does have SEN of some kind, then you may find that a good state school can access a wider variety of the help that he will need than a private one may do.