Is it worth pursuing a refusal to assess for my autistic son's EHCP application

Obvious reasons why it might not be granted is that he is achieving well academically in school, good attendance and no behavioural concerns. However, he has to wear a significantly adapted uniform which takes a significant emotional toil on him to wear. He would be a school refuser were there to be a small shift in circumstances for example my husband works in the school and this may not always be the case, certainly not in secondary school and his current school have indicated that they may change the policy on his adapted uniform. We are considering getting an educational pscyhologist report (privately funded) now, would this be valuable and worth the investment for this process or to support ongoing adjustments during his remaining educational years. He is in a private school currently so we have also thought about requesting an EHCP (no funding) route. There is no local authority involvement in his care currently but he is likely to move into the state sector in secondary school. He is in year 5.