Cutting dense foam, we used to use a band saw with an old blade with two modifications: - the teeth were hammered flat (ie, no set) and; - the blade was mounted in the saw backwards so the teeth didn't bite into the workpiece.
It was surprisingly effective in a wide variety of materials. And very amusing when someone tried to cut wood when the saw was set up for foam/plastic....
Actually, the larger holes in the foam blade would provide better cooling and reduced friction to prevent melting the foam. But at that price, an experiment with the sheet metal blade would be feasible.