In a dark room, using a high contrast projector with a screen-size that's limiting ftL, going with white over white milliskin might be the better (slightly brighter) choice.
Even white on white is below 1.0gain.
The Sony makes 1100/1400/1700lumens with full-lamp (reference/cinema/gaming) and 750/950/1150lumens with eco-lamp (cinema and reference are accurate while gaming uses a cooler white-balance)..the hw40 doesn't drop much brightness from zooming and both full/eco lamp modes are very quiet.
A 120"-2.39:1 screen will divide those lumens by 50ft-square and a 120"-16:9 screen by 43ft-square at 1.0gain, which gives (cinema+full) 28/33ftL at 120" 2.39/16:9 or (cinema+eco) 19/22ftL at 120" 2.39/16:9. It's safer to assume the spandex is around 0.7gain which brings those down to 20/23ftL (cinema full) or 13/15ftL (cinema eco).
The quiet fans and large brightness difference between eco and full allow the Sony to start at eco and (after several hundred hours) keep brightness high by simply switching to full lamp once if you start to feel like the picture needs it. A dark-colored, dark room will also go a long ways for allowing a dimmer image to appear brighter and bolder than the ftL might suggest. Many large theatres stretch lamp-hours until brightness is down to 12ftL, and you'll be well above that for practically the entire life of your lamp while being able to use a room with less light (theatres have fire and trip safety mandates) and a spandex screen will show less artifacts than a perforated or MFG weaved screen.
If you fear that you may be more of a brightness enthusiast, is there any possibility of moving the seating closer (8 or 10ft away) and dropping down to an 8ft-wide screen?