Authentication

If I've understood correctly then Moodle can't use this database for authenticating users. You could still use it to manage users and their profile data. You could then manage a process to change affected users' passwords, perhaps using Password format = "Internal" in Moodle. But if the database is being used for authentication with other systems then this may be an issue as users would have one password for Moodle and another for other systems.

Another option could be to change Moodle adding support for this password format. You'd need to edit two files, first auth/db/settings.php to add a new $passtype[] element (line 98 in Moodle 3.8):

    $passtype["mysqlfunction"]    = "MySQL Function";

Then edit auth/db/auth.php to handle this new type (line 141):

            } else if ($this->config->passtype === 'mysqlfunction') {
                $authdb = $this->db_init();
                $rs = $authdb->Execute("SELECT PASSWORD('$extpassword') AS password");
                $password_function = $rs->fields['password'];
                $authdb->Close();
                return ($fromdb === $password_function);
            } else {

This would add "MySQL Function" to the plugin's Password format field. It seems to work correctly in my tests.

Caveats:

• Modifying Moodle isn't generally recommended, changes may delay updates being installed and the changes must be re-applied as part of the update.
• When you modify Moodle it's more difficult to get support, either in these forums or elsewhere, as your Moodle doesn't match anyone else's.
  
• MySQL's PASSWORD() has been deprecated for some time and has been removed from MySQL 8.0 so ideally plan to move away from this regardless of what you do.
  

But this change may be an option for you in the short term.