I think you misunderstand what sort of "work" children benefit from at home. NOT "supplementary" school work. What is the point? School work is for school. The role of the parent is to enrich their child's education by conversations, experiences and interests OUTSIDE of the curriculum. Or to reinforce the curriculum content by practical applications BEHIND the place they currently are at school.
Why would you be working with your child on academic work beyond what he is covering at school? Where is the pleasure or the value in that? There is none.
Children benefit from shopping, talking, political engagement, travel, camping, gardening, cooking, visiting museums, galleries, exhibitions, voluntary work, hobbies, languages, music, conservation, sport, hiking, trips and trip planning, sewing, craft, community service, films, photography, family life with relatives of all ages, etc etc etc
What they don't benefit from is doing more school work at home with parents, beyond where they are at school. By all means work on weaknesses and support with problem areas, a little bit, in a non threatening way.
But you keep saying your son is 2-3 years ahead - not unusual in state schools, and now you are saying you have taught him at home, so that 2-3 years ( which you have assessed in a home setting, where he will inevitably perform better) is not down to natural ability, but to cramming, so I take back what I said earlier about him possibly being in the top 10%. He might be, but there is no evidence of that. Maybe top third? Might not be grammar school material.
Nothing you have said makes it likely that your child would qualify for any sort of funding for a private school.
I know you want the best for your child, but I am still advising that supporting him through state school with a varied and interesting programme of family activities and hobbies outside of school, plus academic support in weak, behind areas, is the way to provide it.