I guess the container was less important in how I was thinking about it. A free floating piece of ice is never fully submerged without some outside force. The density of the ice will determine how much of it is above the surface of the water which is also how the balance of the ice cube displacing the same amount of water as its liquid equivalent would. When the water goes from solid to liquid, its density increases which means the volume it displaces decreases.
School Iceberg Experiment
If the iceberg actually extends all the way to the sea floor, then it is being propped up and that would certainly change what happens as it melts. I would imagine this is the issue we are facing here and probably what you were alluding to by stacking ice on top of what would free float in the glass.
Yeah. I had to go back and edit my post. I originally said it wouldn't cause the water level to rise, but my fingers typed faster than my brain worked.