Is this an SOA where important data might lie in another service or data store, or is this a monolithic app and DB that is responsible for many (or all) things?
What was stored in that database? Does this affect user data? Code?
Regarding what's gone: The production PostgreSQL database. This suggests that the code itself is fine, but the mappings to the users are gone. But git is a distributed VCS after all, so all the code should be on the developer's machines as well.
Have gitlab considered something like ZFS snapshots? They don't have the consistency problems or the space wasting and reliability problems that LVM brings. You could have it take snapshots every few minutes.