I seem to recall that contact cement and styrofoam is a very bad combination. The solvents in the glue just melt the styrofoam almost instantly. I’d test a bit of the insulation if I were you. There is a product called “liquid nails” which might be appropriate? It’s fairly tough and robust.
I am with Tony on this one, use “No more nails” or “hot glue gun” both will work equally well.