The wire in Hot wire foam cutters can move around, for one reason or another. I use mine on Polystyrene for insulation of my Shed. The best foam cutter and most accurate is a wire about 1/16" thick with very small sharp protrusions, the wire spins and is extremely taught. It's a professional machine and quite large, but does extremely accurate fine contours.
I mention this as someone might be able to copy this idea on a smaller scale. Quite obviously the wire must be available commercially.
Hope that helps.
George