Create a Macro. Tools, Macro, Record New Macro. When recording, do a "Save As" command and select each location in turn. Stop Macro.
When you run the Macro, you will be prompted to overwrite each saved spreadsheet. You could create a button to run the Macro. You might be able to create some vba code to run the Macro on closing the spreadsheet. Sorry, pushed for time so this is all I can give you!
I have created a macro , not easy for somebody with little knowledge of Excel, now I can save the file in various locations, not with one but a few clicks, much easier than how I was doing it before.
I asked Google how to create a macro.
Boxtops it can be done not quite as we wanted but close enough