OP - wow does this sound exactly like my now 10 year old DS. What your child is experiencing now is exactly what happened to my DS at the parochial school he was attending for kindergarten.
One thing you need to know is that his mainstream private does not have to accommodate your son, and many will not especially if the accommodations are a disruption to other students. The school will slowly force you out by constantly reprimanding him, pulling him out of class - and then you will come to the conclusion that you don't want your DS in that environment and you will pull him out. That's pretty much how it goes down. That way their hands are clean.
First, Who diagnosed your DS? was it a psychologist?
Is he medicated? I know no one wants to medicate and we too were in total denial with regards to meds but it was either that or continue to watch our DS spiral downwards, lose friends, not access the curriculum because he cant concentrate etc. You get the picture. It is something you might want to give serious consideration to.
Do you live in a strong public school district? I would consider that route, and perhaps get him on an IEP. Many times as kids get older their symptoms get worse before they improve.
with regards to non-mainstream privates, no one will take a child who is even mildly disruptive

Yes, even SN schools.