I don't know about such a product, too. But I can give you idea for a workaround:

What can be done is to

 1. deny move/delete permissions to files in the library
 2. set up a [request tracker][1] (ticketing tool, there are many: [(1)][2], [(2)][3]) where users have to fill a request (ticket) for deletion on renaming. After admin's approval, the action can be done *automatically*. I mean, if request form is filled correctly, there is not a big problem to issue necessary move/delete command based on its data. If you make entering requests simple enough by removing unnecessary default form fields (severity, priority, subject, etc.), then this might be very easy to use because the entire form could have only two fields:

        Full Path of File:   [...............................]
        New File Name*:      [...............................]
           *enter new file name with or without path OR enter 'd' to delete the file
                             [Submit]  [Cancel]
  
 **Tip:** Users can be taught one simple trick to **copy full path** of desired file into the form:  
 If the file is clicked with <kbd>Shift</kbd>+<i>Right Click</i>, its context menu reveals ***Copy as path*** command. Then user can instantly paste full path of the file into request form – no typing needed.


  [1]: http://en.wikipedia.org/wiki/Request_Tracker
  [2]: http://en.wikipedia.org/wiki/Comparison_of_help_desk_issue_tracking_software
  [3]: http://en.wikipedia.org/wiki/Comparison_of_issue_tracking_systems