I've started using ewm to get this kind of unification between emacs window management and non-emacs window management.
Lagging is super annoying, and regretfully it comes mostly from very slow pgtk cairo implementation.
I'm getting the Skia branch to a usable shape this week. A bit too early to say, but the approach looks promising. Running it for almost a week with a 4k 120hz external monitor, no issues noticed so far.
I switched to Wayland and sway about 2 years ago and it hasn't been bad, but this is the thing that makes me most sad about Wayland. X has so many different window manager options, it's like, an embarrassment of wealth. Wayland has like, 3 "finished" ones (if sway even counts).