Start clock or not???

[Illini_Ref]

Here is the situation. 4th down, team A has the ball and a six point lead. 45 seconds left for the game, clock running. Clock gets to 23 seconds and Team A commits a falsse start. Official stops the clock for the penalty, enforces it, and puts the ball back ready for play. DO YOU START THE CLOCK and let Team A run the clock out???

I know that in most instances you start the clock, but how about in this case. I know there is a rule that lets the referee start or stop the clock if he feels a team is gaining an advantage and manipulating the clock.

I don't think I wuld wind the clock unitl the snap in this case. Opinions please.