You probably want to bind-mount in some kind of /etc/mailname that actually makes sense. Probably also worth setting -h to the same value. Something like -v /etc/mailname:/etc/mailname:ro -h "$(cat /etc/mailname)" might be reasonable.
If you want to use this with another container that has sSMTP or similar installed, here's a configuration you can adapt: (assuming your second container uses something like --link some-exim4:smtp)
Mailhub=smtp
FromLineOverride=Yes
Then sendmail in your linked container should work as expected. You can also skip FromLineOverride in the second container if you want, but the alternative is sSMTP being weird about hostnames and forcing you into a box.
If you'd rather not relay mail directly (which is a smart thing to not want to do generally), you can trivially configure this container to relay via a Gmail account instead! Just add -e GMAIL_USER=youruser@yourdomain.com -e GMAIL_PASSWORD=yourpasswordhere and the entrypoint will automatically preconfigure to relay via the Gmail account specified!