According to this https://forum.godotengine.org/t/switching-the-running-game-to-use-wayland/68394/... it should just work by adding the `--display-driver wayland` flag, given that a recent enough version of Godot was used to export the game.

Unfortunately that flag does not appear in `--help` and seems to be ignored when used. 
Seeing how the above post is from June and refers to a Godot 4.3 dev 3 snapshot the version you use is probably not yet recent enough. I would assume that once that trickles down into stable it should just work.

EDIT: The game works perfectly fine on Linux. It's just that it still uses Xorg, or well, Xwayland on a wayland compositor instead of running natively on wayland.