To get a statement you only need evidence from a professional of what his needs are. You also need to show that the school cannot meet those needs from their existing resources.
It isn't easy to get one by any means and the LA will try usually try to say that the school can meet the child's needs by making certain adjustments so a statement isn't necessary.
However my DD attends a specialist Autism unit which can only be accessed by children with a diagnosis.
Why are you so worried about professionals attributing his difficulties to a named disorder? My DD1 has found a much greater understanding of herself since she had her diagnosis. She feels that her difficulties make more sense and that she has something in common with a specific group of people rather than just feeling like a weird misfit as she did before.
Is there something you think a label would do to your son that supporting him without a diagnosis wouldn't?
My LEA publish criteria for each area. The most difficult one to meet is probably cognition and learning where a child would have to be just a L1 going into Y7 to qualify.
That kind of blanket policy is illegal and will no doubt be challenged at tribunal or judicial review at some point. Children's needs and the school's ability to meet them must be assessed on an individual basis.