When my son was twelve, and first went to his special school, the male and female TAs used to do mentoring work, for typically 6-8 hours per week, being paid by Direct Payments.
There are people who can do the work. There is some cross-over where TAs in special schools, might also work as support workers in residential accommodation for adults with SEN and LD, and some of those will have worked with adults with ASD, SEN and LD in MH wards if hospitals. It may seem off-putting now, when your child is twelve, to think about staff who work in residential provisions and hospitals, but actually, as your kids get to be older teenagers with ASD, and bigger, it is comforting to know that there is a group of workers who have the compassion understanding and skills to work with autism.
When LAs help young adults with ASD and SEN to live independently, they often engage mentors to go out with them and support them - so these people are in the workforce, although, if you don't know where to look for them, you wouldn't realise . These people can work somewhat flexibly with teenagers, too.
By aged twelve, most kids, who would have had child minders before, are independent, which makes it hard for we parents to find carers, but there is a way of doing it, if you have a helpful LA. Lots of people want to do this caring job, in London. I can't speak for other parts of the country, but these stsff are prepared to travel a certain distance to work. Staff who work in PRUs also do this support work at weekends and after school.
I would ask advice from your nearest special school.