I've been using Lutris on i3 with multiple monitors. Enabling gamescope has been extremely helpful. I just get a window that tells the game to render at some fixed resolution, and it is happy to scale fullscreen or any size. I've been using i3's fullscreen toggle, it goes to the current monitor, or I can move it to another with my regular i3 binds.

It looks like gamescope has it's own fullscreen shortcut (super + f), but people complain about it about it going to the wrong display. Maybe your window manager offers a more consistent full screen option like i3 does.