In my experience in dealing with this often, a TV feed is treated as being part of Hard FX and not BGz. So in that regard usually we always build from the ground up, then comp it down into 1 or two tracks, even for military/riot/crime scene stuff. On some occasions, shows have played a visual on a TV in which they obtained permission to use the footage, and that "found footage" has an audio track which is then provided in the OMF. From there we just sweeten what's already there with a light FX pass. Had to do that once for about 20 minutes worth of footage across a show where we're in a house repeadtedly that has monster truck sports footage playing on a TV in the BG - cut FX for all the monster truck revs (match when we see onscreen, cheat OS ones around dialogue), along with a steady crowd bed and spotted rise/falls of crowd cheers and jeers.
Usually TV feeds play so low in the mix that if for some reason the source audio has to be dropped and only the FX played, it would sell just fine. I personally wouldn't sweat it too hard, unless the TV has a larger story purpose and/or is featured prominently (e.g. ECU shots of the screen).
Besides, I always find something fun about creating a movie within a movie ;)