Hi,
Glad if can help you 
In case of run a script is not a straight effort to do it by clicking an icon.
However it could be this way:
- script is placed in Indesign Startup Folder;
- there is an user interaction window inside a script body to determine if InDesign should quit at the end of process;
- desktop shortcut icon is linked to InDesign app.
...If you still can see a benefits with opening ID app just to copy some files...
rgds
Jarek