I assume you consciously changed your copy command to the copyright, unless you meant Alt+Ctrl+C, which is the default copyright symbol in later versions of Word. In any case, how about creating an Autotext for the degree symbol? Type the symbol and copy it, open Autocorrect options, choose something like dg/ for your shortcut and type it into Replace box, and paste the symbol into the Replace With box.

Or, turn your keyboard NumLock on and type 0176 to get the symbol. (I find that cumbersome.)