I wish I could understand this as what you say still does not make sense to me. The fact that water freezes at a lower temperature under pressure some how does not explain why the suspended object has an increase in potential energy. I did not say it was an especially big object as anything suspended right at the water survice would be lifted and therefore have a net increase in potential energy. I would suspect that the same is true if you could support all the ice that forms above the water surface. Of course this increase would be offset by a decrease in the remaining water/ice.
Oh here is how I see it although I know my logic has to be flawed.
Energy of object = X
Energy of Water = Y
Energy of Cooling substance = Z
Energy needed to freeze water = F
Total energy of all these = E
So E = X + Y + Z
So as Y and Z exchange energy I would see it as
E = X + (Y-F) + (Z+F)
But as the water freezes the potential energy of X would increase. So where is this come from. I just cannot see how taking energy away from water would increase its ability to do work (lifting object). Even if it take a lower temperature to freeze the flow of energy is away from the water. I am not questioning any of the answers above, I am just trying to understand what you are saying.