I think we have a slightly different understanding of the state of athletic department budgets, due in part to how purposefully opaque they are... yes, many athletic departments barely "break even," but my understanding is that is often a somewhat artificial consequence of them purposefully making sure to use every last penny to maintain that appearance, often on inconsequential endeavors like slight modifications to facilities. I say the above acknowledging wholeheartedly that that could be entirely wrong, or (more likely) could be true for some but not all institutions... I looked into this more deeply when I was at The Chronicle, but that was a long time ago now.
But let's go with your premise that all the money in existing TV deals is accounted for. How's this for a creative, reasonable solution: past college athletes get paid "royalties" just like actors do for syndicated shows and movies. BTN wants to replay the great Michigan v. Michigan State game where Braylon Edwards carried Michigan to an OT victory? Great, but
on top of their current deal with the conference, there must be a new deal that compensates the players who participated in that game. BTN wants to use a clip of Denard Robinson in a promo? Same deal. In all likelihood those sums would be much more reasonable, just like royalties in other entertainment venues are, but it would come from the network itself rather than the athletic department.
I think that might thread the needle, no?