This might give you hope to know what we have here and with that knowledge you might be able to access a private school place where you are.
We have a large independent international school. The school takes from 9 months to 18 years. We have 4 locations, the school is huge! The maximum classes in R-Y2 is 16. The maximum class size for Y3+ is 22.
Firstly every child gets 15 hours funding until the the term after they turn 5 (So, if born before Christmas funding for the first term, before Easter funding for 2 terms and before 31st August funding all 3 terms). But, if they join Reception at state school, parents can only use that funding for afterschool clubs and childminders/nannies. It really helps for pre-prep schools because it is discounted from the bill. My eldest got full year funding discount off the Reception bill because she was born in July. My son got funding for 1 term (November), my second daughter got funding for 2 terms (February).
Now, the school my eldest daughter went to is very much prominent in our town and in fact, the county. They open up their facilities for swimming, sports, science clubs at the weekend etc etc to both local schools and the community.
The reason is that they want to expand the school and it's facilities so they have opened up what they have at a discount so the community can use what they have. This allows the council to approve their planning applications. It's a sensible strategy.
The one thing that is universal though, although not openly publicised is that private schools must make available places for children not usually able to access private education. That could be accepting diversity, providing bursaries and equal scholarships.
What I would do if I was you is approach the school you think is best suited to manage your son's needs and look at class sizes and ask them what a 1-2-1 would cost. For our largest school that was £1000 per month back in 2014, so that might have changed a bit since then. I would then ask about a fees bursary which for our school is 40%, plus for his first year you also get the under 5 funding for however many terms (do plan ahead for the next 2 years as the bill will go up a bit). There are other schemes that the school can signpost you to, speak to the finance department.
The big difference is that state schools are very restricted in their offerings, whereas private schools can do whatever they want. At state schools, it is very much stand in line and do as you are told so we can cope with 30 energetic children. At independent schools, if a child is a bit hyperactive or is having a difficult time coping with a task, they can change it or take them outside to let off some steam on a whim, it's their choice.
It is worth noting though, that not all independent schools are the same, some won't budge and some will move mountains, you just have to go in and have a feel for the schools individually. Get a plan in writing and it will help if you can write a plan in a risk assessment style yourself as you know your son and what works or doesn't work.
Risk - Has a short attention span and is likely to be overactive
Effect - Disruption to other students and teaching style
Solution - Fidget toys, sports orientated curriculum, break-out space
*Notes - X is interested in tractors and loves doing tasks surrounding gardening.
Once the school has accepted your son and a coping plan, you can then go to the LA and fight for the funding. If they say no, bat it back to them and ask them if they can evidence a better provision. They won't be able to and you can force them to fund him at your chosen school.
We have 3 good private schools in our county town and no fewer than 9 - yes 9!! Private schools in our county. It is not just because we are a affluent county, it's because the state schools (especially the secondary schools) are not great. I hate that we pay twice for education, we shouldn't have to, state schools should be able to choose how they use their funding and allow more openness to their teaching methods.