I don't think that I have the right to get a label for anyone
Why do you consider a medical diagnosis of autism to be a label?
Labels are things that you stick on jars of jam or suitcases. Medical diagnoses open doors to support.
School have suggested he's assessed, that's likely to be because he needs funding for the support they wnt to give him, without a dignosis and the extra funding that brings, they may not be able to meet his needs.
Caveat, we all know this shouldn't be the case, but very often it is.
particularly if it changes the opportunities that life can offer
What changes would those be?
The ONLY professions that will not at present accept someone with a diagnosis of Autism are the Armed Services.
The Police welcome autistic applicants and even have their own Autism Association www.npaa.org.uk/
STEM, Academia and IT have a high percentage of neurodiverse employees.
I think that is an adult decision for him to make about him. Not me
You have already received first hand accounts on this thread from autistic people whose life would have been made so much easier if they had been given a childhod diagnosis which explained WHY they were different to their peers. I'd suggest you research this further on other sites so you can understand what a vast difference early diagnosis can make to an autistic person's life.
IF he is autistic, he was born autistic and will be autistic all of his life. No-one can change that and no-one can change him. He is who he is.
You have the opportunity to find out whether he is autistic and IF he is, to have strategies put into place to support his individual needs.
If he is not autistic, he will not receive diagnosis.
Why don't you go down the assessment route and tell the team that you believe he has PTSD, not autism and ask them to do screenings for both?
You'll need to be prepared for the possibility that he may actually have both.