Could you not just use a trigger somewhere at the cave entrance to reduce the skylights intensity to a value, where it is dark enough in your cave, or turn it off completely, if necessary? Or use triggers to switch/blend between two different skylights, one for your daytime outside world, and one for dark places like your cave.
The easiest would be, if there is a trigger, that does not switch from 100% to 0% instantly, but let the change happen gradually, maybe depending on distance, how far away you are to the bright side or the dark side of this trigger.