How can I display iPad content fullscreen with HDMI on a TV?
There are apps that detect if your iPad is mirrored to an external device via HDMI and adjust to fit accordingly.  For example, the official Netflix app outputs the original aspect ratio to the external screen.  The device displays the message "Now playing on your TV" as noted in the attached screen shot.  The TV displays the show at the proper aspect ratio depending on its settings.

Keynote and other apps work similarly.  It would depend largely on the aspect ratio of the TV show or Movie being played.