Is it possible somehow to add this automatically to either the familyName and/or the fileName (for instance when adding those in a Custom parameter)
Best solution for this would probably be your own export script. For an example, look at what Nico Hagenburger did in his repository.
Okay, thank you!
Or a script to update the version number and familyName (and/or fileName) together and use that instead of manually updating the version number.
You could write a Filter plugin that does just this.