Okay, it works – for now! 
Thanks to everyone who helped me so far.
Just one thing: Currently I’m using $SNAP_USER_DATA for my user files. It seems to work just fine on my local installation.
What exactly are the bugs which make most developers use $SNAP_USER_COMMON instead?
EDIT: Is it this one? https://forum.snapcraft.io/t/bug-saves-are-blocked-to-snap-user-data-if-snap-updates-when-it-is-already-running