I ended using a script that sends incremented version to firebase:
It was possible to setup pubspec.yaml in such a way it syncs with the iOS and android app version and build version: How to set build and version number of Flutter app
After that it's possible to setup a custom build script reading and incrementing version like given in this GitHub issue: https://github.com/flutter/flutter/issues/41955
This makes sure that version gets incremented
# Find and increment the version number.
perl -i -pe 's/^(version:\s+\d+\.\d+\.)(\d+)(\+)(\d+)$/$1.($2+1).$3.($4+1)/e' pubspec.yaml
...Coming from the GitHub thread