If you already have a MovieClip
with instance name startMenu
placed on the stage, so no need for,
private var startMenu:MovieClip;
you remove this from your code.
And always have the stage instance first and then proceed.
So modify your constructor like so:
public function mcStartGameScreen()
{
addEventListener(Event.ADDED_TO_STAGE, onAddedToStage);
}
private function onAddedToStage(e:Event):void
{
startMenu.mcStart.buttonMode = true; //Now this will not give the ERROR!
startMenu.mcStart.addEventListener(MouseEvent.CLICK, startOnClick);
}