I wonder if what you need is not an “unsporty” school - because I think they are all more sporty than you’d like - but a school where they are willing to consider your concerns and your DS’s needs and to accommodate that to an extent at least.
If you can get a formal dyspraxia diagnosis that may help, but that said I suspect a lot of
PE/games/SEN professionals may take the view that an appropriately tailored sports programme (rather than doing as little sport as possible) would be really beneficial for him.
In addition, even if your DS struggles with and doesn’t enjoy sport, there is a risk as he gets into the secondary years that he won’t want to feel different from his peers.
I would try to contact SEN and heads of sport at the schools you’re interested in, share your concerns and ask how they would approach this as I’m sure it’s something they’ve dealt with many times before.
When we were looking at schools I was adamant that my DS would not play contact rugby, due to concerns about head injuries. I was prepared to find a sympathetic neurologist to say he shouldn’t play if necessary! I contacted the head of sport at St Albans about it and got a very long and thoughtful response, the upshot of which was that they even though all boys had to play rugby in the first term, they do offer non-contact rugby, and that’s what my son has done.
When he started the school we went to a talk about the sports programme, part of which was extolling the virtues of contact rugby. However, this was again very thoughtfully done, and was absolutely not a hard-sell. So I was left feeling that they did respect parents’ views, and were not pushing a “one size fits all”.
In addition, the school runs a club called “Fun Fit Fridays” which is (as the name suggests!) a fun, casual way to take part in sports. Pupils can choose whatever they fancy - table tennis, badminton, swimming (nerf gun battles?) and join in for as long as they want, or not bother at all if they don’t feel like it that week. My DS and his not-very-sporty friends all love this.
So I think a good school will want to be inclusive towards your DS by offering him sporting opportunities which suit him, well beyond the competitive rugby etc option.