Setting doubleclickenabled won't help in this case, because that only works for textfields under iOS. You can just use a MouseEvent.CLICK listener, and make a note of the getTimer(). If the new click is less than a certain amount of time since the last one, then it's a double tap. You could go further and make sure that a third tap doesn't happen too soon, otherwise the kids could just frantically tap on the button to get through.
Here's something useful that someone made to measure any given number of taps:
https://github.com/fljot/Gestouch
With that you could add a listener like this:
var doubleTap:TapGesture = new TapGesture(myButton);
doubleTap.numTapsRequired = 2;
doubleTap.addEventListener(GestureEvent.GESTURE_RECOGNIZED, onDoubleTap);
...
private function onDoubleTap(event:GestureEvent):void
{
// handle double tap!