A lot of board games come with little hourglasses. You could use one of those as a prop to help time it. Stick with 10 rounds but if the PC doesn't tell you what they are doing before the timer is up you move to the next one?
Just a thought.
A 10-minute hourglass sounds like a perfect tool for this encounter. Not only do have the "hurry up aspect" of a live countdown timer, but the bottom of the hourglass filling with sand would be a perfect visual for how much the room has filled up. (In this case you wouldn't run things in combat rounds, but simply quickly go around the table asking each character what they were doing. If the player can't answer in a few seconds, their PC is momentarily frozen with panic and you move on to the next.)