My son has a similar profile to yours but with the addition of being very far behind academically, although very bright in a non-academic way. He attends an independent mainstream primary funded by the local authority. Although it is technically not a specialist setting they have a very high proportion of SEN children and a very nurturing approach with small classes.
It's not something I set out to try to achieve. It's more where we landed after years in mainstream state primary where every day was a struggle, in spite of having an EHCP. When he was on a reduced timetable at school, with the LA paying ££££ on alternative provision to plug the gaps, I convinced his mainstream state primary to finally put in writing that they couldn't meet his needs. We then consulted with various specialist placements who all, quite rightly I think, said that they didn't think they could meet his needs either, but it felt like that needed to happen if only to rule them out. There was only one option left and that was the independent school that I had also asked to be included in the consultation and that just happened to be brilliant for him.
In addition, the fees for the mainstream independent school, even with a bit extra added on top for some extra support, would cost the local authority less than some of the specialist schools, and they are certainly cheaper than EOTAS, which is what I had asked the local authority for, but it feels like you need to fail in far more than one setting before they will agree to that. So in the end the independent was a bit of no-brainer. While it didn't feel like much of an administrative fight, we certainly went through the mill in being in the wrong school for too long..
Apart from needing to secure an EHCP, you should really look into what is out there in terms of the different types of specialist schools before you rule them out. I agree with what others have said, in that there are many types of specialist school. Some of them are very academic and pupils are expected to be performing at least to age-related expectations in order to even be considered. You may find one to be a better fit than the independent school you have your eye on.