Not telling the school and pediatrician is a HUGE mistake. You're not hiding it from the school- they already know (hence the identification and IEP), they're just not allowed to tell you what they suspect the diagnosis may be. I promise you the Kindergarten teacher, speech therapist, OT and school psychologist will figure it out within a month. I know my team always did when I was a school psychologist! It's what we're trained to do.
As for the pediatrician... again... silly. Why don't you want her to know? This is your child's primary care physician and she should have a complete record of his medical history.
Hiding things doesn't end well and your private psychologist doesn't have a crystal ball and I am very sorry that he or she has led you to believe that your son will be "cured" in 3 years. It's not ethical to do so. Your son doesn't have a disease to be cured, he has a syndrome which by definition is a cluster of behaviors.
This IS on his medical records- the psychologist keeps records, the OT keeps records, the speech therapist keeps records.
I can only tell you from the "other side of the desk"... parents think they are hiding a diagnosis, but in all reality we figure it out pretty damn quick and at some point you will be asked by the school if you have had your child privately evaluated b/c they have "noticed some behaviors". Do you intend to lie? Not a good path to head down...
Your child will progress much faster if ALL clinicians working with him (in school and privately) are working together towards common goals, not "undoing" eachothers work unknowingly.