Can I make a subdirectory in public:// viewable in Media module's UI?
I guess you can five a try to filedepot module if it helps you. Here is a quote about it (from the module's project page):


  ...  full featured Document Management module that has a google docs like feel. It fulfills the need for an integrated file management module supporting role and user based security. Documents can be saved outside the Drupal public directory to protect documents for safe access and distribution.