I seem to remember those hot wire tools that could cut expanded polystyrene - as I remember they gave it a nice smooth surface but I don't think it made it in any way stronger so I don't think your plan would work.
I once had success mending a fridge door shelf with Araldite using some kind of mould to shape it as I needed but I can't remember for the life of me how I stopped it sticking to the mould!