Is it required for the worshipper to remain in the mosque until asked to leave, in order to validate their reward for staying for remembrance with the intention (of doing so) when the mosque closes after Fajr prayer?

As long as the reason for your departure from the mosque was the impossibility of sitting in it due to its closure by the person in charge, then your departure at that point, without him explicitly ordering you to leave, does not affect the validity of your intention or the reward associated with it.