Cloning an Exception (kinda sorta)

I would actually go the route getting those developers to follow good practice. As a whole, the field would be better off if developers like that were mentored rather than something like this.
 
So, I would recommend using either of the techniques you put forth originally (using the Data property or a specialized exception).
 
Granted, if you're working in legacy code, it isn't always possible, but it is the ideal. :)