That looks like my handwriting. I managed my GCSEs, a level and degree somehow (I tried my best and it would be better at the start of the exam but be worse than this at the end).
Then I was doing professional exams that involved writing 15+ sides of paper. I was pulled aside by a tutor and told I was going to miss out on marks because the examiner wasn't going to be able to pick up on all my arguements. As I didn't have dyslexia or any other identified LD or neurodiversity diagnosed, I wasn't able to use a computer. So they suggested I write in capitals. I can now write in very legible capitals almost as fast as my drunk spider scrawl. It might look like I'm shouting but at least other people (and me!) can read it.
If the school are concerned, can you ask them for a solution beyond 'just improve your handwriting'. With any luck they'll suggest laptop, or capitals, or printing and as it has come from the school and is therefore THE RULES and not mummy who can't possibly know what she's talking about (I don't have a child with ASD oh no not me) maybe he'll accept that he is allowed to deviate in this instance?