It would be nice to have launch options, similar to Steam, but they would only be useful for apps like Butler and not in the website itself.
As a workaround, maybe you can make extra files with custom arguments. So if your executable is called my_game, you can make a bash file my_game_fullscreen.sh that looks like this:
#! /bin/sh
./my_game –fullscreen
I’m not familiar how Butler works, so not sure if there are some extra details needed to make this work.