Let's be clear. The BD screen is a solution to a poor room, it is not the solution to a good room. A good room doesn't need a screen to correct for ambient light and reflections because a good room won't have those issues to begin with. So the BD screen is the solution to $100 in paint, some blackout shades, and dark carpet. Which, very well is worth it for a multipurpose room like the one you have.
So, I would recommend, if you are going 100% projection, which I would do, then the BD screen is appropriate for your space and you should not spend so much money on a more expensive projector as that really doesn't make sense. A great projector is what you use in a great home theater, and you do not have a great home theater, you have a nice multipurpose room, but it's a long way from a home theater, and shouldn't be treated as one. It should be treated as what it is, and your experience will be maximized by remembering that you want the best image in the space you have.
The BD screen, which is pricey, may seem to be a silly screen to use with a lesser expensive projector, but it is well worth it IMO for the use and the space. A BenQ W1500 on a BD screen. The largest fixed BD screen available is 113" and in your space, that's my biggest hangup. The screen is smaller than I would recommend. It's appropriate for a 12' viewing distance to feel 'center of theater', but it is going to actually appear small to people playing pool. Since I'm spending your money, I would probably put local displays around the pool table, etc. if I were doing it. In fact, when I finish my basement at some point, that's probably exactly what I will end up doing. But, that's up to you. If the basement is not built, and you are just looking at plans/pre-drywall stuff, then I would really 'dream' of options, and run wires everywhere you want them.
I said before, you should consider the implications when putting speakers in ceiling if there are living spaces above the speakers as sound will bleed upwards into the rooms and floors.
I would not get the zero-edge BD screen, but their standard fixed frame with the 1.4 material.
TBF113BD14 - 113" diagonal, 16:9 aspect ratio, Black Diamond, 1.4 gain, fixed frame screen
MSRP: $3,299 - Perhaps $2,500 or so online (you can PM me)
So, while a projector like the JVC-RS46 or one of the Sony LCoS projectors is the 'best' projector in a theater, it is not the best projector outside of the theater. They aren't bright enough to deal with ambient light, and may get hurt overall by the BD screens that suck up light and hurt viewing angles. I haven't, in all fairness, used the BD screen in an open space like you have, but I have used the DNP Supernova with a theater projector in a brighter room and was extremely disappointed with the results. Blacks were great, but whites had absolutely no 'pop' to them at all. So, I would hesitate to recommend a projector which can't put up some decent light output on your 113" screen.
I also would think the lower price tag would help let you evaluate performance for a few years (or more) and then let you decide later on if you want something brighter, with better black levels, or perhaps both. That way you don't spend a lot now on a projector which doesn't perfectly meet your needs, and then wish for more later after already investing a bunch of cash. No, spend the money on a good screen now, then buy a lesser expensive projector. Use it for a few years and see if there is anything specific which you aren't happy about, then upgrade that aspect later on.
Be aware, the BD screens need the projector to be a certain distance away from them at a certain height for best results. Otherwise, they will reject the light from the projector itself.
Oh, and you still want to ensure you aren't running any lights anywhere which do fall directly on the screen if you can avoid it.
