I read an article on this subject on an ACC-focused site and something stuck with me. It was a small thing at the time and I've since been unable to find that page (wish I could give credit). Why not allow teams to stage a single exhibition game against an FCS opponent? You align the games in about the same time frame as the spring intra-squad games and they can be televised to whatever extent the conferences and networks like.
It solves any of the problems that come from playing a 13th in-season game or adding an exhibition game to the current 12-game season. We all know that voters will use a game in their decision-making process regardless of being exhibition or not, and there's no extension to the length of the season. And, these games would be of greater value to the teams themselves as opportunities to see their newest players on the field in that environment and provide a mid-spring evaluation point for players and units as a whole.
Win, win. No?