It depends on what this application uses the database for.

If you're using the database for system-wide configuration of the module, then putting the database with the module makes some sense, but I get the impression that this is not the case.

If the application uses a single database for all the users of the database, then sqlite isn't really an appropriate choice. For full-blown databases, they'll be running as a different process which will have access to the database files and that problem goes away.

If the application uses a separate database for each user, then sqlite might be appropriate, but you should allow the user to specify the database location themselves via command-line switch or environment variable or config-file or ...

Since perl has DBI, unless you're using sqlite-specific features, I'd seriously consider simply letting the user specify database connection info (dsn, user, password, attrs) and, again, the problem goes away.