I had to do something like this several years ago. I believe I just asked the person requesting the authentication for his e-mail, went into the FAA Airmen Certification On-Line service, clicked "Request Verification of Airmen Privledges" and put their e-mail address into the "Other E-mail" box. The addressee gets an e-mail with the attachment of a signed letter in .pdf format directly from the FAA Civil Aviation Registry within minutes. I believe some of the other overly-official requests for certified and stamped letters (etc.) are because most civil aviation authorities don't have such a user-friendly system (or they want to charge $$$ for the letter).
Airmen Certification - Airmen On-Line Services
Try this option first to see if your airline will accept it.