What is it that makes you believe that that could be a special order serial number?
The 3566 limited was a Lew Horton distributor exclusive. Smith and Wesson records would have has zero entries as to who any of the end purchasers were, with the possible exceptions of a few that went to factory employees and perhaps folks that filled out warranty cards or had repairs
Lew Horton has been out of business for many years, the person who has possession of company records would be able to tell you how many came into Lew Horton and what FFL they shipped to. LH would not be able to tell you who they were ultimately sold to.
Additionally if they did have the information to whom they were eventually sold to, they would be hesitant to give it out to you.
There are privacy issues at stake here.
I know that if I purchased a firearm and I got a phone call from someone who bought it used down the road wanting to know if I bought the serial number next to it or anything else to go with it or if I could find the box or grips, I would be very upset at whomever disclose that information