My DS has ASD and Statement. The DX was a battle and took 4 years. We got the statement at the end of year 6. Now DS is in 6 form. Without the statement he would not even be at school. It was absolutely instrumental for my DS. Although, perhaps, my DS is a more "severe" case than yours.
All the same, SN provision is a process. You need to follow the yellow brick road.
I would suggest concidering carefully what MarianneSolong said, and getting some opinions on the SN children board as well, if you have not already done this, OP.
Is you DS achieving in line with his potential? Are all his needs well understood and fully met? The help your DS receives at school is only going to be helpful if they understand exactly all of the problems and use the right tools. It seems the support you DS is already receiving is based on the assumption that he has ASD. If you DS is "average" is that because of his well documented congnitive tests, or is it because he underperforms without the right help, even though he is getting some?
A lot of DC on the spectrum have problems at the transfer to secondary school and the right support with the right intensity might be crucual. Without the DX, this very much at the school's discretion and if your DS will struggle and underacheive, the school will dismiss that as his average ability.
Yes, a good school would provide the support without the DX, under the speculation that he has ASD, but they may miss a trick, there is also the small question of money and resources. As Wannabestressfree says, this is at the school's discretion and they may schoose which EP advice to ignore if they feel it's "impractical".
I think in the situation of ancertainty you should follow the process. The process would guide you to the right decision when the time is right. First is to identify all of your DS's needs. The school gotten EP and they advised to get a DX- so that is the next step. If DX is confirmed, you will have much more levers and tools at your disposal and the school will have much less discretion..
SN provision is a process and you need to drive it, pushing all the buttons and tightening all the srews. Without the DX you don't even have access to much of this machinery.
Special needs are for life and at every new turn there will be new challenges, having a DX helps to choose and get the tools when needed. It is much harder to get a DX after 13 and even more difficult in adulthood.