As you have an EHCP, I think in your situation William Tyndale sounds like the best bet. They've had an incredible reputation with SEN for a long, long time ie 20+ years. Contact IPSEA for advice as you can get Hackney to pay for transport if you prove it is the only school that can meet his needs. It's important to note that parental preference is written in the law and is just as important as everything else. The only thing you'd have to prove is that the cost of him attending William Tyndale isn't an insufficient use of public funds. Seeing as it's a state school and your other options are also going to require transport, it's unlikely they'd win that argument.
Alternatively I'd look at Grasmere as it's one form entry so nice and small, if also has no uniform so no issues with scratchy polyester. Hackney has a few resource bases for kids with ASD, so that's why schools like Shoreditch Park come up, but unless your child has one of the 10 places, he will be in the mainstream school full time.
You can also name a fully independent school on your child's EHCP, and the LA will have to pay for it. You will most likely need to do this via appeal, but as you're a phase transfer your appeal will be heard quicker. I would suggest you look at specialist private schools that cater to HF autistic children, rather than mainstream independents. They don't usually have very good support in all honesty. My DD is starting at an independent specialist in Sept, and we managed to get it without appeal, so it is possible but you need knowledge on SEND law and a lot of time on your hands. Have a look at IPSEA, it's very helpful.
Is your son starting Reception in Sept 2024?