Going for better school...

Have you taken him to any open days? Many children pick up some very off misconceptions about what private schools are like from their friends. Do bare in mind that scholarships are rarely academic and offer little of any discount and all bursaries are means tested.

A high ability child is not well served going to a mediocre comp. The school's focus will be on their progress 8, that is value added improvement between a child's predicted outcome on entry at year 7 and what they actually get. If a child arrives already likely to be an A or A* student they have no incentive to help them . Poor comp near me raves about one recent student that left to go onto highly regarded Maths degree. Word on the street was that he would have done that anyway, and school just hung onto his coat tails. Top set kids are usually put in the largest classes, with the weakest teachers and not stretched. I agree with Hopping, get him to a taster day/open day/private visit. Y5 kids are wowed in particular by science labs and get the students there to speak to him. Sounds like he'd converse easily with them and benefit loads.

Maybe start by taking him to open days? Once he sees the facilities and, more importantly, has stimulating chats with teachers of his favourite subjects, sees fascinating science displays etc he may decide that he actually WANTS to work to attend a particular school?

Not all school entrance exams need extra coaching. Have an honest chat with the school and see what they say. I did no extra work with my DS and he got a place at a selective with bursary. But the school were adamant that they didn't want coached kids

Unfortunately you can’t make him do the exams etc but I would really really encourage him if at all possible - maybe a taster day at the Private School?
My dd has a part scholarship to a Private School and despite a few “leaving all my friends” wobbles she was always really up for the exam/interview and it’s been absolutely the right decision for her.
She is in touch with a lot of her friends at the local comps and she will regularly come to me and say x happened at that school or y happened at another school, glad I’m not there!!
They say bright kids will be ok at any school but ok isn’t what we wanted for DD and she’s now in an environment where the bright kids are the cool kids instead of being mocked for it

Your DS is, as I recall, exceptionally gifted, not just 'bright' and working way ahead of age-related expectations.

He might not want it at all, but at 10, he is too young to understand the importance of these decisions and a poor school with no sixth form is less likely to be able to accommodate him than a better school with a sixth form (I use my knowledge of teaching A-level to push my brightest students in KS4, for example).

You really do need to be looking into scholarships and private school, or at least beyond a poor local state offering.