Use a web-mailer - set up a script on a web server somewhere and POST a request to it (possibly with authorization), and it'll make an e-mail from it. FF is a browser - use that functionality, and have a server-side script do the heavy lifting.
For the server part, I'd go with PhpMailer, but any mailer script would do.
Manipulating raw sockets from a browser sounds tricky (not to mention platform-dependent); on most systems, you need root/Administrator privileges to do that); also, many ISPs block outgoing SMTP traffic (botnet prevention): SMTP traffic going to a different server than the ISP's SMTP host will get dropped in such case. (e.g. "drop traffic on port 25 to any host except smtp.someisp.example")
As you pointed out, the user has the ISP's SMTP server set up in their mail client, BUT this data is not accessible from inside Firefox; and snooping around in other apps' settings will get your extension flagged as "spyware" (also, how do you find out which mail client is the correct one? My system had two mail clients preinstalled, yet I'm using a different one).