
Valerion
VisionMaster Pro 2
Great picture/gaming, but RBE and no lens shift are dealbreakers.

Running these analyses costs money. Buy through my links to help keep lights on! I may get a small commission.
Not sure what makes a projector good for outside but I just wanted to mention the one I just bought: Optoma UHZ35. High lumen, 4k, and has low refresh rate option (240hz at 1080p) which is why it is marketed as being a projector fit for gaming. It's within your range
I've just bought that uhz35 and it's about 4.9 meters from my wall - it's insanely good. Can count the pores on someone's face on the screen. Love it.
I got the optoma 4k gaming projector. I think its awesome.
End of reviews

Valerion
VisionMaster Pro 2
Great picture/gaming, but RBE and no lens shift are dealbreakers.

Epson
Home Cinema 5050UB (V11H930020)
Great value with huge lens shift, but outdated lamp tech and poor HDR.
Epson
Pro Cinema LS12000 4K PRO-UHD Laser Projector
Superb 4K/120Hz gaming and no RBE, but requires calibration.

BenQ
W1070
Budget king, durable with good image, but RBE can be distracting.

BenQ
HT2060
Excellent 1080p contrast and quiet gaming, but low brightness for big screens.

Ranked #1
Valerion - VisionMaster Pro 2

Ranked #1
Epson - Home Cinema 5050UB (V11H930020)

Ranked #1
Hisense - Laser TV PX3-PRO

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

Ranked #1
BenQ - GV50 Portable Projector

Ranked #1
Anker - Mars 3 Air