At first, I think you should sync your database tables via linux
rsync command. It is a solid way, more as an script inside your wp install and his dependencies.
However you will update the password in the second table after an change of a password. I think you should use the WP hook after the change and fire on this hook an copy of the password. The hook is
* Fires after the user's password is reset.
* @since 4.4.0
* @param object $user The user.
* @param string $new_pass New user password.
do_action( 'after_password_reset', $user, $new_pass );
However you can also use the hook after a change of the profile, that include the change of the password - Hook
* Fires immediately after an existing user is updated.
* @since 2.0.0
* @param int $user_id User ID.
* @param WP_User $old_user_data Object containing user's data prior to update.
do_action( 'profile_update', $user_id, $old_user_data );