If you already use gamemode it has custom section in gamemode.ini file to execute custom scripts before and after game sessions. You simply use gamemoderun %command% as launch option for all of your games.

Another option is directly chaining your scripts with %command% but your mileage may vary.

I was able to log game start and end with this naive launch option.

echo "start" >> ~/Desktop/test.txt && %command% && echo "end" >> ~/Desktop/test.txt