.he is in year 5 and manages without any extra help.
I think you should investigate a statement, but can see some problems, sorry to be the bearer of bad news.
The statement is designed to deliver extra help within the state system, above and beyond the state schools resources.
As your child 'manages' without any extra help it would be hard to prove the need for help, you could argue the next school could only accommodate your child with extra help via a statement. The statement does allow you to name a school and if its selective you only need to pass the exam and not worry about oversubscription criteria.
Talk to your LEA they will have a parent partnership officer who can help.
As a parent of an autistic but very able child who is in school without any support I think you are doing the best for your child, we all hope that one day our children will be independent.
We did apply for a statement at age three it was declined but we did get extra help in a super supported state unit, who took a child who could barely speak and within a year taught him to read.
Good luck on your journey.