I reached out to
archives@tiffany.com to learn more about this service, and unfortunately it looks like a serial number won't be included on any output. From my exchange with them:
"Please see attached for the necessary paperwork to schedule your Commissioned Research Request for your Rolex.
Please note that our Commissioned Research for Watches includes a Certificate of the Research completed, which does not validate the authenticity of the Watch. The Watch Certificate will not include the Serial Number of the watch being researched, but will outline the Watch Title, Model Name, and Reference Number and be accompanied by the Archive’s official seal and signature of our Head Archivist.
If you have any questions or concerns, please do not hesitate to reach back out. Thank you again for your interest in Tiffany & Co.’s history, and have a lovely holiday season.
"