@TheCunkOfPhilomena
I suspect part of the problem here, if not the problem, is that unpredictable, shouty, scary teacher.
I was at an autism conference once and one of the speakers explained that the way we feel when we are running late for a very important meeting is the way people with autism feel all the time.
And then the way we feel when we are stuck in a traffic jam and know we are going to be late for that very important meeting, is the way those with autism feel when something unexpected happens or something changes suddenly, like when this teacher starts shouting.
So your poor little boy goes to school every day in a high level of stress thinking, will she shout today, who will she shout at, will it be him she shouts at. It must be a bit like going somewhere every day where a tiger is chained up in the corner of the room that is fine most of the time, but every so often it has been known to break free and eat someone. Imagine sitting in a room for hours with that tiger wondering if it's going to break free today and eat you. I wouldn't sleep the night before - would you?
And when the teacher starts shouting everyone turns round to look. You know that yourself when someone starts shouting in the street - you turn round to see what's going on. And your son, especially if he has sensory issues, could find that difficult to cope with too.
When my son was a similar age, I wanted to tell his TA about something really good he had done at home. He was horrified. "Oh no! Don't tell Mrs S!" When I asked why not, he explained, "Because she'll tell Mrs B (the teacher) and she'll tell all the children, and they'll all put their eyes on me and that's scary."
So the classroom is a really scary place. No wonder your son wants to stay at home to learn. It's calm, quiet and safe there, and (presumably) nobody shouts.
Apparently not wanting to make mistakes is common in autistic children. My son absolutely hated writing, because when he made a mistake he would have to rub it out and he didn't like the mess it left behind. When the school gave him a laptop to work on he found it so much easier and less stressful.
As you say the SENCO is approachable, I think I would first talk to her, tell her everything that's been happening, and how it's all getting worse and that your son begs to stay at home and is talking of running away from school, and ask what they can put in place to help and support him.
Maybe your son could have a time out space when the stress gets too much and where he could work on his own for a bit. You could ask for an EP to come in to observe and advise what provision might help. If you think it would help, ask about him using a laptop.
I'd also see if you can get a referral to CAMHS or similar. Obviously living in this high state of stress is not sustainable and will likely get worse as he gets older and moves on to secondary school, which is likely to be bigger, louder, busier and more unpredictable than the primary school.
The psychiatrist who assessed my son after a disastrous experience in mainstream FE, told me that the incidence of mental health issues among those with autism is much higher than it is in the general population, so you really need to do all you can to safeguard your son's mental health now.
You may want to consider requesting an EHC Needs assessment. Although your child functions well academically, he has mental, emotional, social communication needs and (possibly?) sensory needs which may need specialist provision to be put in place through an EHC Plan. Information here:
www.ipsea.org.uk/asking-for-an-ehc-needs-assessment
So called "high functioning" children and young people with autism are often very much misunderstood, because they appear to be coping well, but they are not. I know a girl who had a complete nervous breakdown at 15 because she couldn't cope in mainstream school. She was then in and out of education for years before at age 19, she eventually got an EHC Plan and went to a very small specialist placement, where she was finally able to do her A levels.
I would also start looking at secondary provision now (look outside your own local authority as well) as it may be that your son needs a specific mainstream school that has good provision for, and experience of autistic students or even a small specialist autism specific setting, rather than your local school.
Good luck. 