Should my ds be kept in for behaviours he cannot control

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EP report states can't sit still shouts out, no concentration etc
Have fought school for EHCP
Did my own didn't get one
School say they can handle him
Yeah by threatening him with headteacher everyday or keeping him in at break and lunch

what did he do? It's always the same shouting out, not sitting still etc etc
After multiple years of this if I thought he could sit still concentrate etc I'd get him to do it at home
Shouting out random stuff is a stim I believe
I'm so frustrated with the constant punishments it's evident he can't do it so he shouldn't be punished for it?
The worst is though he's a school refuser because punishments are being carried over to next day 'you've lost break tomorrow' etc
I can't keep doing this
I've already moved him once and now he's settled with friends but it's the same old not
believing he's not deliberately being disruptive.

Under investigation for asd/adhd

That sounds like discrimination. He is not violent and while shouting out is annoying and distracting, it is not a huge thing. How old is he? Is this primary?

Punishment will not change his brain chemistry, so what is the point? It will negatively impact his self-esteem, but that's it. They should have a strengths based approach and support him in his weaknesses. What are they actually doing to help him?

If he has ADHD, meds can make a big difference.

Did you appeal against the refusal of an EHC needs assessment? If it was less than two months ago you can still do so. 90% of those appeals succeed.

If not, it could well be worth applying again.

This is standard behaviour policy for our school but my DD with ASD is treated differently and has her own behaviour plan. She does have to leave the class if she's violent or disruptive, and will have it with a TA if this is just before break, but they don't take break away as she really needs it ( in fact she sometimes has extra breaks to help).