My only concern with using a piece of copper with thermal paste on the sides is that the thermal conductivity of copper may not be the same as the original thermal pad. I would assume that it could work, but I would do my best to try and replace the pad with a similar one.
A lot of time, there will be people selling pre-cut slices of thermal pad [http://shop.ebay.com/?_from=R40&_trksid=p3907.m570.l1313&_nkw=thermal+pad&_sacat=See-All-Categories|on ebay].
I have also purchased thermal pads [http://search.digikey.com/scripts/DkSearch/dksus.dll?lang=en&site=US&WT.z_homepage_link=hp_go_button&KeyWords=Thermal+Interface+Material&x=30&y=22|from Digi-Key].
You should be able to read the processor temperature on the desktop. If you do decide to try and use a copper spacer, then be sure to check the temperature and make sure that it is within operating limits.
Good luck and let us know how it works out.