Another possible solution if accuracy is important (your short deadline makes it sounds like it may not be though). If you take tabletop epoxy (the hard, clear stuff designed to do coffee tables and bar tops) or some other clear encapsulation epoxy and let it flow over your foam visor section, you'll have a hard, more thermally stable surface to form your clear styrene sheet over and shouldn't have to spend time recarving a shape you've already carved into a different material. Epoxy over foam is, I believe, how surfboards and stuff are made.
If you can build up a few layers of epoxy, you may even be able to separate it from the foam and use that itself as the visor, but I have no idea if that would work as well as it sounds like it should.
Just a couple of ideas, anyway.