Make sure that you understand the downsides of using a screen like the Black Diamond before you decide to spend that much money. We sell a ton of BD screens, and for good reason. But for a completely light controlled theater I would recommend against it, namely due to the reduced viewing angles and screen artifacts.
That said, if you are intent on going forward with replacing your matte white screen I would recommend the SI Slate. Roughly $3k in a 120" 16:9. You would give up a little light rejection, which you aren't really in need of in the first place.