There doesn't have to be a rod. If the balls allow for fairly flat "wedges" [ < , > ], the cake boards could be heavily hot glued to the surfaces of several balls [all of which are touching].
You could insert dowels into the 2 largest balls and place them into the cake below so that the dowels would be vertical. That would anchor the balls and stabilize them so that they could support the cake above.
That said, I personally think that the two top cakes may be dummies, and therefore extremely light. I say this because of the sharper edges on the fondant.