I'd say HE too.
You can have a statement maintained when you HE - some LAs will try to cease it but legally they have a duty towards children identified as having SENs so it has to be with your agreement. Insist it's kept up so that you can continue to get any supports possible that are laid out in it, plus you may also need it in the future.
When my son was deregistered at age 10 he was a mess, and like you I feared for his mental health if we kept him in the school system - it just didn't work for him and support was minimal, and in fact much of what was provided was damaging to him. He needed a very long time to recover from his experiences and he wasn't able to join even very small HE groups without close supervision for well over a year.
This is why I'd be cautious about staying on a school roll and holding out for LA home tuition. You can't guarantee that tutors allocated to you will have SEN awareness and your son could find that a challenge, and it might even damage him further. Plus, they'd want to provide tuition on their terms, not yours, and if your son's not ready for it because of long-term stress then the LA won't take kindly to it. Also I hear that a lot of home/hospital provision can be erratic and often not worth its while - in some cases it's just LAs ticking boxes to show they're 'providing tuition'.
Also, if you were to pursue a placement at special school at this time, before your son's had a chance to recover, the likelihood of failure is higher. Do you think he could cope with settling into a new school in the next six months? If you think not, then wait awhile till he's more stable.
In your position I'd HE outright so that you can be in control of your son's development, both academic and personal, though you'd still get an annual review of his statement. In practice this could just be a formality to keep it going, and I'd suggest you do that so that it's up to date for when you're ready to return your son to the school system.
I'll admit that I'm speaking from my own experience with a disinterested and unhelpful school and LA and I appreciate that things might be different where you are, but I think it makes sense to proceed with caution.
Whatever you do it seems clear that you can't continue like this as your son is being harmed by a system that isn't right for him, and you need to act to protect him. Best of luck to you all.