How so? The 4 projectors you listed are all worthy of the upgrade, will support your gaming, and will shoot a beautiful picture. You may have to deal with Faux K, but I honestly don't notice a 2K$ difference in PQ that justifies jumping to native 4k, and also once you step up to 120+ (in) immersion is trumps all IMO (but you'd still have a great picture)Small correction to the original post. The room is 16x13 not 16x12. So I'm guessing that I should be able to put up a 120" screen on the wall (where it shows exercise bike in the attached picture)
Going back to the projector discussion. It seems like there isn't a "perfect" projector at least in the price range that I'm looking at (equivalent to 65" OLED). So the compromise would be to either go with a (great 1080p projector or "decent" 4k (pixel shift) projector. From all the reviews that I've read JVC DLAX500R (i think), Epson 5040UB and Optoma UHD65 are the good options but for the price Benq HT2050A seems like a steal.
With that said, since the whole reason to get a projector is to go big and enjoy 4k movies/gaming, I would prefer a 4K projector instead of 1080p but that does not seem to be doable with current projector options......so it's back the drawing board
I'm partial to the Epson since I own it, but my neighbor owns the Optoma and it's also a solid unit. Get a projector and don't look back!