Kevin, I can't thank you enough for your advice. I had an extra PC lying around so I installed SME Server on it. After a lot of configuring, I got it to work in harmony with this printer. The IP address of the server was instantly recognized as a SMTP Gateway.
I set up the SME server in server-only mode. I created email addresses for all the people in the office who want to receive emails from the printer. These addresses are local, but you have the option to forward emails to an outside email address. So, each address is configured to forward to a corresponding account.
This is the solution I've come up with after one day of tinkering. Perhaps it's possible to send email directly to any outside address, but I couldn't figure it out. If I specify an outside address, it fails. But, it's not important. I can live with this solution. It's better than I ever dared to hope for.
Thank you, a thousand times, thank you!