If the teachers are trying to refer you to specialists, that means they see something. And if the teachers are willing to speak up when they see something, then odds are good that you are in a school that is willing to accommodate whatever that something turns out to be.
So, yes, I think you tell them that your son has, in fact, already gone through all that and you have a diagnosis you think fits.
I know it is hard to trust again and go through what is an intense and draining process, but I think you have a better shot this time and will regret it if you don't take it.
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Mom to an amazing young adult AS son, plus an also amazing non-AS daughter. Most likely part of the "Broader Autism Phenotype" (some traits).