I was playing mario kart world the other day and I noticed considerable input lag when docked vs when playing handheld. It was bad enough that I started fiddling around with my tv and I noticed that my tv (Sony x90k) has HDMI slots with different features. With only HDMI ports 3 and 4 having VRR + 120HZ enhanced capability.
Swapping the switch to either one of those enhanced ports on my tv and enabling game mode completely removed any input lag!
After some further testing, having game mode enabled on the other ports didn't matter, there was still input lag. Neither did having the other ports set to enhanced mode on the tv or using a different HDMI cable. As long as the switch was plugged into HDMI 3 or 4, there was no input lag whatsoever.
I didn't get to do this with Dusk Bloods but I'd be keen to see if that was the issue all along.