Epson Home Cinema 3700

y yet I wanted to still get something that was able to look great even with the lights on. So this projector definitely filled all those needs and rig...

I set the lamp to eco mode and lowered the brightness from 50 to about 45 where it started to lose detail, but nothing really did enough to help it. A few other problems also soured the experience, though they could probably have been dealt with if black levels were fine. Most annoying being that that the projector kept losing signal during transitions (menu>loading screen>video), and often not regaining it without stopping and restarting. I assume probably signal confusion from me running it through my old Denon AVR-2310.

I got an Epson 3700 refurb from the Epson store a month ago. It's glorious, and also much cheaper than the refurbed 3800 (the 3700 is the older version of the 3800). I think the main difference is a different version of HDMI, which if memory serves, only matters if you wanna game at 4K 60fps or something. I dunno how it's gonna look after the lamp has hundreds of hours on it, but it's very bright, even on the Cinema + Eco mode.

My knee jerk reaction is the Epson Home Cinema 3100 on the Epson refurb store or the Benq 2050a, if you can find one i stock. These may or may not work with your room and other considerations that you may have. I believe, those to be the best in your price range. You should consider giving details about what your room conditions are from the list in the Wiki.

I would say out of the ones you have listed that the Epson Home Cinema 3800 is the best choice, but I am partial to LCD. Your throw calculations appear to be correct for the Epson at least (the only one I checked). From 12 ft with the Epson you can go screen size between 77 - 125 inches.