
Valerion
VisionMaster Pro 2
Great picture, good gaming, but pronounced RBE and no lens shift.

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I just got the BenQ TK700. It’s below your budget but checks a lot of the boxes. Super low input lag. Great for gaming. Great picture. It’s mostly white but has a grey front panel.
First, that simulated view is crazy realistic. What did you use to do it? For the projector, I use a BENQ TK700; it has great brightness, color and 240HZ with 4ms input lag. I use it with a grey screen to make up for some of the light that's reflected off the walls.
For gaming, you are generally looking at DLP models for extremely fast response time and low input lag on many models (but not all, so check!). You are typically working with lamp-based models at this price, and you can get factory refurbs of models like the BenQ TK700 (not the BenQ TK700STi which is short throw) for that money. JMGO was (is?) having a sale which puts one of their 4K models at $1,000, which is a reasonably good model for the money as well. What else may be on the used market is a tough call as there are a ton of 'used' projectors out there at varying price points. Optoma's UHD series have had very poor reliability, so that's something to consider as well in your search, but some people have had great luck with their models. Your choice on whether to consider them or not.
Go BenQ TK700 if gaming is priority low lag, view sonic PX701-4K if movies matter more better colors value. XGIMI lifestyle less for serious gaming.
Ceiling mounted with a throw distance of around 13ft 4in, 150inch screen. I'm not a projector expert, learning from folks here as I read posts. But I used the projector for a lot of 4k gaming, and during daytime TK700 was really struggling with contrast with dark scenes. I have blackout curtains but sometimes too lazy to draw all of them, since the setup is in the living room with many windows. With ht4550i, contrast is much better. When I upgraded, the TK700 only had 500 lamp hours, so no degradation. Other than the contrast, TK700 was pretty good in general, but the lens shift, build in Android TV, and faster starting time of ht4550i are also very nice to have.
If you have ambient light, TK700 is not going to be super great with dark scenes in movies. Ht4550i will be slightly better, but can't compare with TVs. At night without any ambient light, both are great.
I don’t think that budget gets you into gaming 4k. That’s basically base model 4k. You can get a used Benq tk700 with extended warranty

Valerion
VisionMaster Pro 2
Great picture, good gaming, but pronounced RBE and no lens shift.

Epson
Home Cinema 5050UB (V11H930020)
Best value, flexible lens shift, but dated tech, poor HDR.
Epson
Pro Cinema LS12000 4K PRO-UHD Laser Projector
Great 4K/120Hz gaming, no RBE, but needs calibration.

JVC
DLA-RS1200 / DLA-NZ500
Unmatched black levels for movies, poor for gaming.

BenQ
HT2060
Great 1080p gaming, deep blacks, but low brightness.

Ranked #1
Valerion - VisionMaster Pro 2
Ranked #1
Epson - Pro Cinema LS12000 4K PRO-UHD Laser Projector

Ranked #1
Hisense - Laser TV PX3-PRO

Ranked #1
Anker - NEBULA X1 4K Triple Laser Projector (D2351)

Ranked #1
Anker - NEBULA X1 4K Triple Laser Projector (D2351)

Ranked #1
JMGO - PicoFlix