I was just wondering if you mounted a curtain rod (black, if you can find one), that controls both sides with one pull of a cord for manual control would work. Then make some curtains that fits the screen area out of some black felt material, or whatever works for this.
Edit: It looks like they are called Traverse Rods.Here's
one option in black. You could always glue black felt to the front of it, to make it non-reflective.