Whyyyyy do people think it's OK to say "is he an only child?" as if this might explain everything!?!? Folks, it's not a guarantee of an unruly, spoilt, ungrateful child. Come on!
Yet I can generally spot one within three lessons. Also fairly accurate on youngest in the family or who has a massive age gap. I don't know how, it's just quite clear having taught for so long.
OP, I worked in an inclusive school with a lot of children on the spectrum for many years and I agree with you. The sheer terror of being seen to break rules is very typical of some of the children I've worked with and goes a long way to explain why he's a lot more polite at school. Two weeks of unstructured time at home with different things happening daily, even hourly, will be absolutely the most difficult for him.
If rules work for him, put some in place. Chunk down why you want him to do. Give him the star chart with clear expectations and reasons for getting stars. Whilst I like the clicker idea, I've never seen it used with children, only dogs and horses! Children can speak and reason, so while it's a fine idea, perhaps to the verbal route first.
I think it will be very hard to maintain a very structured environment at home, but if you can, put some things in place for him which must be adhered to and reduce screen time, it's far too easy to allow children on the spectrum to withdraw into devices, this really needs limiting, IMO. Talk to school and see where he excels, the SENCO should be able to give you some good ideas on how to cope with his time at home. Do please post on the SEND board.