I'm experiencing the same difficulties with SQLite as an Altium database. The database file can't be updated while Altium is open and had created a connection to the database. There are two workarounds I've found so far:
The default database journal style, DELETE, creates a copy of the database, temporarily, writes to the database file, and, when the write is verified, deletes the temporary copy.
Write ahead logging, WAL, puts all changes in a temporary file while leaving the original copy of the database alone. Processes accessing the database parse the database file and the WAL temporary files, and return information reflecting all changes. I can confirm that writes outside of Altium are seen when doing a database refresh within Altium.
The writes stored in the WAL file are written to the original database when a checkpoint operation happens. Either when a process calls PRAGMA wal_checkpoint;
or after a number of operations specified by the WAL Auto Checkpoint setting.
You can enable WAL journal mode in DB Browser's Edit Pragmas tab:
DB Browser Database Settings
The problem still isn't fixed, though. Checkpointing the database doesn't work while Altium is open. The database file is still locked from changes.
The caveat for us is that we want to backup and maintain a history of the database file in Git (yes, this is not a recommended use of Git). Writes to the journal files aren't tracked in Git, which doesn't see a change in the database file until it is checkpointed. This means that we can't push or pull from Git while Altium is open. We could track the database and journal files, but the other consequence is that a simple checkpoint with no data changes will be seen by Git as a change.
Use an alternative ODBC driver
I tested an alternative, proprietary ODBC driver from devart. When I configure Altium to connect to the SQLite database using the devart driver, DB Browser has no problems writing to the file while Altium is open, regardless of journal settings. Altium can also see those changes with a refresh operation.
Conclusion
Christian Werner's ODBC driver (as of version 0.9996) is locking the file in a manner that the devart driver does not, even while Atlium maintains a connection to the database through the driver. WAL journal mode can be a workaround for some situations.