BSPWM, laptop has 1920x1080 resolution, external screen has 1680x1050.
xrandr --output DP-1 1680x1050 cuts image, if I change resolution to 1920x1080, I get "full" image, but quality is just bad.
I have another machine with XFCE, it's display utility does the trick and I wonder can I achieve smth similar on BSPWM with some simple console command?
I use i3-wm and just set my laptop display and external monitor to their native modes manually with xrandr. Been doing it this way for years without an issue. The only time I've seen the output get chopped like you mention is with mirroring, where you have to use the lowest common mode - but I don't mirror, I set each display independently as a separate output for i3 (but on the same X DISPLAY).
I also don't use a login manager, I login to a VTY and startx, old school but simple and reliable.
I also don't mirror, just switch from laptop to external and vice versa. Strange because lighdtm doesn't break anything for XFCE, quite interesting what is the reason
maybe I'm just one of very few that have external resolution lower than host
maybe I'm just one of very few that have external resolution lower than host
You aren't.
How do you have your displays set up? Your xrandr command didn't seem complete. You say you aren't mirroring; are you then using both the laptop and external monitor at the same time? If so, I'd expect a command more like:
Ah, I thought you were displaying on both outputs, not switching between them, hence my mirroring comment. I suspect XFCE, not the DM, detects the output change and takes care of it. You might need to emulate that behavior with a hook of some type that you have to setup yourself with the tiling WM, and you might have to --off the unused display. I'd be willing to bet you can find some sort of hook script out there that can do this, I seem to recall an autorandr program I used in the past where you could set up output profiles. I hope that helps, maybe a little bit.