4K projector with low input lag for FPS gaming?

Current setup:

-Dedicated dark room (blackout curtains with very low ambient light)

- 120" Screen

- Optima HD 72 projector: https://www.projectorcentral.com/Optoma-HD72.htm

- Recently purchased PS4 Pro and would like to start using 4K, but projector won't support 4k.

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I've tried looking into this and my top 2 options are either the Optima UHD60 (lag 56ms) or the BenQ5550 (58ms).

https://www.projectorcentral.com/best-selling-home-theater-projectors.htm

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From what I've read, the more complex the image, the greater the input lag. For example, if a image is 1080p with input lag of ~60, and you increase the resolution to 2160p (4K) then the input lag will also increase. This sounds plausible as the lag is the delay on generating image, and if you up the image quality further it will take longer to process.

This concerns me if I'm wanting to play 4k FPS games.

I wasn't sure if this would even be much of an issue, as I had never checked the input lag of the Optima 72. It isn't something I ever noticed (honestly, never considered). So, I tried to check the lag on the projector I currently have. On a 1080p image, it had a lag of 20-29 milliseconds (didn't check in the gaming setting, which I never used). I didn't get an exact number because my camera wasn't good enough to capture the last digit.

Lastly, the majority of the games I play are FPS. However, I'm not someone trying to go pro. When I read about input lag it frequently references people that are "serious FPS gamers". I'm not sure if I fall in this group (I'm never in the strongest group on the map), but if you look at my gaming selection it's almost exclusively FPS. Thus, I don't know if their definition of "serious FPS" says the best of the best FPS players will notice input lag vs people who play a lot of FPS will notice input lag.

Any thoughts or recommendations are most welcome.

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Any suggestions?