If the Amp is set to passthrough then it should add no or very negligible lag. While I'm not a FPS gamer I've played twitchy stuff on my 4k screen from my Pro through my Amp and adjusting TV settings from Game to Standard made far more difference than anything the amp may had added.
You just need to be sure that whatever you buy will pass all of the things you want. Just passing 4k isn't enough if you also want HDR. You might not have an HDR TV now of course, but if you think you might in the next 3 or 4 years then make sure the amp passes it.
I have a Pioneer VSX-831, in home audio terms it's cheap and cheerful at under £300 and does me just fine. 6 inputs and 1 output.
Sadly I don't think there are many reviews of Amps that measure lag as they do for TV reviews.
Another option for you if you literally just want to have all things at 4k is to use a 4k HDMI switch but I'm not sure how many are out there right now.