Well -- here is how I did it ... for future readers.
I set the sendmail_path in php.ini to /usr/sbin/sendmail -t -i on the web server
I set the "Smart" relay host in /etc/mail/sendmail.cf to: DS<myip-or-fqdn> on the web server
I set my /etc/tcp.smtp to 127.allow,RELAYCLIENT="<myip-or-fqdn>" on the mail server
Restarted sendmail and qmail on the respective servers.
Everything works.