You don't even have to show the television's screen, if you play it right.
Shoot towards people from television's direction... maybe see the back of the tv in the foreground, even. Or if shooting so the tv is totally out of frame, you can use some christmas lights (or something fancier) and a foil-covered bounceboard to reflect & move the "light from the tv" onto the people's faces. In both instances, you can get away with just using a voice-over or the audio.
The dvd idea is cool, too. Obviously a little more pre-production required to pull that off. (Make sure camera's running at same hz of the tv screen, to avoid those rumbly lines)