I tried to reproduce it this morning. While I couldn't play the whole game free, I think I'm on to something. If you press the green X guide button right after you enter the final 'demo' room but before the trial message pops up, then dismiss the 'guide' menu, you can keep playing your game even though the trial message is on top of you. So, it seems the Xbox guide menus override the game ones.
It's possible the game I was downloading in the background completed right when I entered that final demo room thus displaying the 'completed downloading' message, which might override the trial timeout message. Maybe there's another way to force a similar message (like a friend logging onto Xbox live).