
What a bunch of fannies we are. There is a very simple solution to this and we are all overlooking the elephant in the room.
Macros!
For an example:
1) Create a new document, call it "test" and save it somewhere.
2) Go to [tools]>[macro]>[record a new macro]
3) In the dialog, give the macro a name, let's call it "multisave" (note: the name cannot contain spaces).
4) Once you have chosen a name the dialog should disappear with icons for recording macros appearing.
Depending on the version of word, or your specific word setting this could be in the toolbar or floating free in the document. In my Word 2003, it looks like..

You should also note that the cursor has changed, so it now shows a little cassette icon below it. If neither the icon or the toolbar appears, check your help file (press [F1] to bring it up) and read up about macros and how to start them.
5) Once you have the toolbar icons, you are recording. Don't worry it won't record mouse movements only actions. So what we are going to do is save the document as a backup, then save it back to it's original location.
6) Go to [file]>[save as..] and save the file to a new location (let's call it "test-backup"), then do the process again only saving as back to the original location. The reason you do this twice is that "save as" moves the location of the default document to the position it was saved in, so doing a "save as" over the original location will stop any confusion over whether you are editing the original or the backup.
Note: You will need to OK the prompt asking if you wish to overwrite the original document. This only happens when you do this process manually. Once it's recorded as a macro you won't get that warning.
7) Once you have recorded the macro (and the macro itself only contains two actions, both "save as..") click the 'Stop Recording' icon on the toolbar (mine is shown as a blue square).
That's it, you've completed your macro. You will see that the icon on the cursor is back to normal and possibly that the macro toolbar has disappeared.
To prove this works, close the document. If it asks you to save then go ahead, although the macro is associated with the default template (normal.dot) so any new document you create will have the macro available to it (obviously when starting a new document, you might want to amend the macro to point to a relevant backup file for the document).
Now from scratch (ie: no Word open), double click over your original document. Make some amendments to the text, then instead of clicking save as normal, you would do this..
[tools]>[macro]>[macros]
From there a dialog will show up listing all the macros on your system. Hopefully our "multisave" one will be listed, if not you need to repeat the steps above. If it is, highlight it, then click on [run]. No prompt will tell you it's run, but no error will suggest it has. And that should be it, you have saved the latest document in two different places.
To prove this has happened; close down word again, then open up the "test-backup" doc, rather than the original and you will see that the amendment you made to the original was saved as the backup.
If you have any problems with that, let me know.
Edit: I am also being seriously lazy as this wouldn't take more than a couple of hours to create a word plugin to do the same job without the hassle of the macros. Might do that later.