Then I'm not quite sure what I've done because the 'Nintendo 3DS' folder in the root of my SD card is around 18GB
Even with this, there's enough spare room on the SD to create the necessary temporary files to copy stuff off, so I don't really understand where the issue with it all is.
Thank you for the suggestion, I'll check it out if I can't resolve what's going on at the source.
From what you said where you copying to does it have enough space? If you copying 18GB of data you need 18GB of space on destination.