I went back and tried again (Outlook Express), and after some dinking I discovered I had to turn on outgoing authentication for FP's SMTP server. Once that was turned on then FP's SMTP server would accept something other than @myfairpoint.net in the sender's e-mail field. Thanks for prodding me take another look.
Of course, it would have been nice if the FP tech that I spoke with on Saturday knew that answer ...