Oh I empathise. I have an autistic kiddo very high needs and they can't find a school that can meet need and when they do I'm sure he will refuse to go after a few days. Also have an older kid and a toddler.
I was at breaking point. Still kind of am but now have a disability so I worker who put pressure on the LA and now I have 2 x 2.5 hour breaks from my SEND kiddo (half hour drive there and back and it fucks up toddler nap but I get to have a shower and eat chocolate or do a toddler group).
Your son gets HRC so you qualify for a disability social worker. They probably will say you don't but put in a formal complaint and ou will get one in the end. Honestly our has done very little but what he has done has saved my sanity there is absolutely no support otherwise. You can probably get respite for your disabled child if you push for it.
Sign your school refuser up to young carers. Once a term my non disabled child goes to one of their events for a couple hours. It's not much but it's something. Also Cubs/scouts isn't that expensive and once a week you get time to do a food shop then they do lots of camps where you get a whole day or two off!! My SEND child can't manage it but if you think your school refuser could I seriously recommend.
Legally if your child is out of school for 15 days (they don't have to be consecutive) the LA must provide alternative provision. This will likely be a tutor but you can push for alternative provision. If your child is refusing school the school cannot meet need. Join the FB group EOTAS matters- it's a massive fight to get alternative education but it can be done with repeatedly making formal complaints and quoting the law. Join the group as there is a wealth of advice in there on how to get an education your son can access. Obviously the focus is on education not giving you a break but it amounts to the same thing.
Also do you have a local FB for parents of disabled kids? Look for one and ask advise in there. You will get signposted to local stuff and local places to go. Often they have meet ups that you can take your kid to and let them play while you get o have a cuppa and a chat and feel like a person no just an exhausted carer.
Also do you know about EHCPs? To be a school refuser there is likely unmet educational needs. Apply for one and th support your kid gets in school may change his attitude to attending. You will get turned down for the assessment but you speak and then they will do the assessment and say no don't need support and you appeal- this process takes over a year but over 90% of appeals win. As part of the assessment you will have educational psychologist assessment possibly speech and language and OT (I thought SaLT would say my kid was fine as his speech is great but it came back he's got really bad processing speeds and can't hold much in his head when working stuff out or hold the questions in his head long enough to formulate an answer. I had no idea. And this is a massive reason he refuses school when he does have a place). Anyway with those reports if they find needs you can try applying for DLA and then I'd use that money for a regular babysitter.
There is absolutely no support for families of disabled children unless you fight for it and formally complain loads. We are all fighting over pathetic ly funded resources. It's disgusting because we have to find the energy to sort this stuff at the times we are feeling most desperate.