I think this whole area of "compulsory" post 16 education and young people who have disabilities, illness and/or special needs is a complete mess. Sorry, I know that's not helpful and may not be what you want to here but I think you might need to be prepared.
Whereas, with an under 16, the LA has to find a school, once your dc is post 16 there is just a mish mash of provision that they are expected to slot into.
In my area there are no state schools with 6th forms attached so you have to go to college. Consequently I don't know if the school is allowed to refuse your son a place but no doubt someone else will be along with more information on that. Could your ds not take GCSE exams for one year in the lower 6th and then go on to do A levels? Schools used to offer their students a repeat year like this when I was young (in the dim and distant past), does that not happen anymore?
Colleges don't seem to offer complete GCSE repeat years . They have to offer GCSE in English and Maths, and I suspect most do Science but other subjects may not be covered except those that fit into the level 2 Btec category (so ones that can be considered more vocational than academic).
If your dc cannot make the attendance requirements of some college courses due to ill health -(in my ds's case a mix of physical and mental factors) you can be just taken off the course. They basically ignored my ds's needs anyway. My complaint about this is still ongoing.
Courses/schemes for dc not at college tend to be aimed at "work skills" and "basic English and Maths". So my ds, who despite missing the last 7 months of school had managed to get 5 GCSE (including English and Maths) at C+ could have been signed up for a level one course, which I'm assuming would be also highly inappropriate for your ds too as he'd be already working above that level. Or there are very physical "outward bound" type courses for the "disaffected" and so on. So if you are bright but are not well enough to do an apprenticeship or college course there is nothing out there. And home education is only "allowed" for child benefit purposes if your dc started prior to 16.
Sorry, I know this is grim. Hopefully, school will come through for your ds and it may be that your local colleges are more flexible/helpful than our local one.