Hi David,
Instead of lighting the foam core, buy or make a large scrim (diffusion material stretched on a frame). I've done tons of these kinds of subjects over the years, and I've found that it's much easier control light coming through a surface rather than bouncing off of one.
You can feather the light in infinite variations-- hard grad, soft grad, etc. Then, for small flaws that still remain, fix them in post.
I remember one time in particular I was shooting floor lamps for a department store, and had to light a silver sphere that reflected everything. It took a long time, because I had to set up multiple scrims and reflectors, and there was always some small flaw to address. If only we had digital back then!
-john