As the owner of 2 637 Wyatts, I would also be interested to know.
If I remember correctly, the first widely demonstrated "Wyatt" model was finished within minutes on camera at S&W HQ with a stock 637 and reviewed, test fired on the spot. This was "reality" TV, so who knows what really happened during that meeting?
For me, the internal and shrouded hammer models do have a different feel/balance ...and I've come to prefer the bobbed 637.
Did S&W simply have an excess of 637s on hand, waiting for a project such as this? Pure conjecture on my part.
Sorry, just my opinions rather than the answer you sought.
Happy shooting and hope you enjoy it.