You are not forced to pay for private school. Your DS has an absolute right to educational provision, in common with all children. The LA must fund independent placement if they can't place in a state mainstream or specialist school - and they are also lying about that, because they can force a mainstream to take him, even if that mainstream is full. If a school is named on the child's EHCP then that school has a duty to admit.
I'm afraid state specialists are thousands of places short across the country, now. Some are many kids over their intended legal limit, because more and more are placed there via Tribunals. It's a crying scandal and the harm it's doing everyone (because kids who need to be there are being placed in mainstreams: if the class teacher and TA are being diverted to manage extremely high needs, how are they going to focus properly on the other kids?) is likely, I suspect, to be the subject of a public inquiry in a few years from now. And that's not even considering the kids who, like yours, are told there is no space and oh dear, that's a shame. I know - me, personally - dozens of families whose kids have been out of school for a long time, some of them years. When the Covid lockdowns happened, and all the parents squawked about OMG the lost months in education, my eyes rolled so hard I felt like I was in the Exorcist. Lack of provision is close to normal, for kids like ours, at this point.
We are now in a situation where no parent of a disabled child should, or can, rely on their LA to do the right thing. You need to upskill and learn about their rights, and the law, and become their best advocate, or they will be left in the dust. So, so many kids are at home with nothing.
The only school that can block a disabled child being placed with them is an independent school - and not all independent schools, either, as charities with 'non-maintained special school' status, or special schools on a list known as Section 41, can't block a child either. The status of the school determines what the legal rights are. It goes:
Maintained school (whether Academy or Free School or not): LA can name it on an EHCP and the school must admit
Maintained special school (whether Academy or Free School or not): LA can name it on an EHCP and the school must admit
Section 41 special school: an independent special school that has applied to be on a preferred list. These schools must admit in the same way as a state one.
Non-maintained special school (usually managed under a charitable trust of some kind) and they must admit, just like the others above.
Independent school (whether mainstream or specialist) the LA CANNOT name it, and placement can only be with their agreement.
The status of the school is set out on their entry on the gov't schools info page. You can run a search for all schools, of all types, within a sensible distance of your home (for a small child this should be around a 40 min drive, and an hour for an older one - the LA will have to fund school transport if it is unreasonable to expect a child of that age and with those needs to walk it, though be warned most LAs ignore that being the law for disabled kids and apply the standard 2 miles below 8, 3 miles above it criteria before free school transport is awarded). You can also see the school status - type of school - on the entry, and if it is section 41.
The LA have the power to name any school other than one that is genuinely independent, and so does a Tribunal judge.
The legal routes to make the LA do something are Tribunals, or, in very specific and fairly limited circumstances, via pre-action letter for Judicial Review (parents have to pay for the letter, but if the LA do not concede and the case is strong enough, it can then be continued in the name of the child, who will get legal aid - hence the need for the case to be strong enough as it is scrutinised for merit before the funding is granted). Refusal to name any school at all, on the part of the LA, would likely meet that criteria.
IPSEA and SOSSEN offer good free legal advice.
There is no such thing as cheap legal advice once you are in the realms of its being paid for, but there are also some excellent solicitors and direct access barristers out there who do amazing work securing disabled children their rights. Happy to pass along names if you need that, and want to PM me.
You have also done brilliantly to get a child this small a strong EHCP. But the point of that EHCP is to protect his rights - which the LA are failing to do, because making you pay for private school is so much cheaper for them.
We are in a national SEN crisis right now and the only kids getting anything worth having seem to be those with parents who won't accept less. Good luck, and I am so sorry you're in the trenches with the rest of us here.