/home/lutris/.config is owned by root
Copying PulseAudio socket causes the /home/lutris/.config directory to be owned by root, thus Lutris cannot write to /home/lutris/.config/lutris.
This causes Lutris to fail during startup.
The issue is not visible if the Docker volume is already created, so perhaps this might be a regression?
I believe you meant auth token of PulseAudio? Issue might be the creation of this directory https://github.com/Nicceboy/gamify-containers/blob/de0bfb1e32c54a3162cc8145c8be8cc65744a16b/entrypoint.sh#L35:
Volume mount takes recursive ownership of $HOME directory only on the second run for this folder (this folder is created after first), so at the first time might need to set correct ownership as well.