I FOUND THE SOLUTION!
Here's the trick to getting your old backed up save file to work:
1. Find the title ID of your game (it starts with 0004).
2. On your SD card, search for this title ID. Normally there will be one result, a folder. Inside this folder will be a file called 00000001.sav. Back it up (heh) to your computer for now.
3. Still inside this folder with the backed up save file, move up three levels until you're in a folder with a long, random name. Now, open the folder called 'title', then open whichever folder corresponds to the first half of your game's title ID, for example, '00040000'. Now, open the folder that corresponds with the second half of your game's title ID, for example, '000fc600'. Then, open 'data'.
4. Inside this folder you will find another file called 00000001.sav. Delete it.
5. Now, copy the backed up save file you backed up to your computer to this folder.
6. Load up the Checkpoint save manager (install it if you don't have it already).
7. Look for your game in the list, press A to select it, press L to back up your save file, just use the default name.
8. When it's done backing up (which takes only a second), move your cursor to select the backup you just created, and press R to restore.
9. Load up your game, and continue where you left off on your old save file.