I thought this might be the case too, but I tested this quickly by duplicating a set in my set lists. The global part count went up by the total number of pieces in the set, plus the set's spare parts.
On a side note, I think I can leverage the custom list functionality after all. All I need to do is copy the spare parts into a custom list, then keep the list open in a separate browser tab when looking for replacement lost pieces. I think this will work quite well, albeit knowing to look for the spare parts in the custom list.