Disclaimer – never done it, myself.
You’ll likely never get what you’re looking for by trying to fold it in to a 5.1 mix, as you’ve found out there is not enough channel separation. You’ll need an authoring program which gives you the capability for a second audio track. This is a track completely independent of the primary 5.1 mix, often used for additional language and commentary tracks. The DVD spec allows up to 8 total tracks, each of which could be 5.1, pcm or whatever. Don’t know how or if DVD studio supports this.
Sounds like you want something else to chase the DVD while preserving nice DVD audio as well. I suppose the ability to send one audio track to perhaps the optical output, and the other (tc) audio track to analog output would be player-dependent. Maybe some DVD players would do this, I have no idea.
However, with timecode chase you could always have your nice audio on an external device for playback. Then, TC could be on the primary audio track of the DVD.
Is there a video engineer responsible for the show controller who can give you more guidance? Have you seen this done before?