I'd start the assessment process particularly because of his slow processing speed and his inability to keep up with his peers since starting year 4.
It can take a couple of years to get to dx, so the earlier you start, the earlier you may have interventions in place for him. If you have school's SENCO on board, all the better.
If you can get SENCO to write a letter to confirm the Y4 difficulties including processing speed, handwriting, inattention etc.and his sport coach to write a letter confirming what you said he's done and why, then also mention diagnosed ASD in the family, that will hopefully help the GP see the need for a dx.
We all know that school should provide interventions for his needs whether he has a dx or not, but not all schools do, so it's wise to start the path to dx now.
I wouldn't hazard a guess as to what a dx could be, conditions run co-morbidly so ASD, ADHD, ADD, SPD, PDA, Dyslexia Dyspraxia and a few others are not necessarily stand-alone, that's why dx is often done by a team.
Be prepared for long delays and to have to justify why you want an assessment, keep a note of any behaviours that are different to his peers and hang on in there. It's a bumpy ride but you'll get there eventually 