You need both, and it does not matter which one comes first, but it may be better if you get the private results first, espcially since he is high functioning.
Call a Developmental Pedatrician and get the next appointment, which will be many months out. Also, write, do not call, the local school district and request a meeting to grant your consent to evaluate your child becuase you suspect that he has a disablity.
School distrcits educate children, doctors treat childres medical needs. Autism is a medical condition that creates educational needs. The school cannot diagnose autism, all they can do is identify educational needs assoicated with or consitent with a particular diagnosis. If the school district finds that your son needs services, they will still need a doctor to sign off on his diagnosis.
You have the wrong worry. The school district is not out to lable your child, they are more likely to take one look at him, and tell you that they dont see a darn thing going on. You need not fear a lable at all, there is nothing about a diagnosis that will hurt him, and you will find yourself fighting to get recognized, not to keep them from excluding him...this is a very common unfounded fear. The thing you have to fear is that your son will need educational services and get none, until he starts to get into trouble or fails a state assessment, and then, and only then, will they listen to you that something is going on and your son needs help. Special ed is not a place, it is a service, and you will be spending the rest of his educational days arguing that he needs more than they are ever willing to give him.
Log on to www.wrightslaw.com. Scroll down the left side of the page. Click on retention. Read about it. You need to get your son to kindergarten on time. He does not need to be older than his class, he needs to get as much intervention as he can get as early as he can get it, and he won't get that if some of his delays are masked by his age. Holding a child with needs back is one of the most devistatingly destructive things you can do.
Read about advocacy on wrightslaw. There is much you need to learn about this process. If it feels overwhelming, find your states yellow pages on the site, and find an educational advocate to help you. It is much different than you think.
M.