The density change in the liquid water due to heating / cooling over the likely temperature range is tiny compared to the result of going from ice to liquid wat

The density change in the liquid water due to heating / cooling over the likely temperature range is tiny compared to the result of going from ice to liquid water. But I'll award you pedantry points nonetheless :) The precise outcome requires knowing the volumes and shapes of the ice and water components, their initial temperatures and the temperature of the ambient environment so that you can calculate the path of the water volume over time as the ice has melts. In order to make that calculation you'd still have to make use of Archimedes Principle in order to know that you didn't need to worry about how much water was displaced by the floating ice in your calculations, which was the original point.