Oh lol. I think the clock starts after you earn your first trophy. Theoretically I could complete the entire Fury campaign without a gold medal in anything and not once barrel rolling and not get any trophies. Then it would just take getting the last medal to start the clock which makes a 12 hour clear possible. Do you need gold medals for the campaign ships, or just a medal in every cell?
Edit: I don't think silver in every cell is enough to move onto the next set of cells. Damn, that would have been a clever strategy. If I'm careful I can still grind for loyalty points before starting the clock though.