I've worked as a Speech and Drama teacher at private schools. You tend to get access to more extra curricular activities at private school. I coul help with confidence and the voice and how to use it, but I'm not a Speech Therapist... And private schools do not have resident Speech Therapists.
Yes, you'll have smaller classes... And that might help with his confidence.. and give him a more softer experience with his stutter/confidence. But, I don't believe a private school will give him access to anything more than a state school .
It sounds like you need reassurance.. And you need to speak to the school now, so they can tell you exactly what they can do for your son.
Do it now, and it will really help you and your son prepare for September.
If I were you, send him to private school. I have 4 children, so private isn't an option. But, if I was in your situation, and find finances could be sorted.. Do it. But, insist in asking the school for support with your son's stutter/confidence and what they can put in place for him.
Private elocution/speech and drama would be really great for him.. And if its a good school, they should have a resident teacher.
It could be one to one, two to one or, normally in reception years, it's larger groups of say 4-6 kids. But, you can request smaller, you'll just pay a bit more. It would go hand in hand with his Speech Therapy.. I would give him fun mouth exercises, teach him short poems and I would, overtime, prepare him to enter Speech and Drama exams, like LAMDA, and would maybe pair him up with someone and enter them at a local Competitive Festival to recite a fun poem together with some fun movement. Over time, this is all possible.
Arrange a meeting at the school and find out what they can offer. Put your mind at rest. x