I have a 3DS that came with a game pre-loaded on it. How would I go about deleting the game (NOT the data if possible) off of the system to free up space? And since I never actually received a game code or cartridge, how would I go about re-downloading that game?
Yes, you are able to delete a pre-loaded game from your 3DS (as it is treated as a eShop purchase).
You can redownload it as any other eShop game that you have purchased in the past.
You need to be logged in a Nintendo Network ID account so the game is linked, since you need a NNID to be able to enter eShop, that souldn't be an issue.
Here you can read further in detail.
Once deleted, pre-installed software (Mario Kart 7, Swapnote, Fire Emblem: Awakening) and Nintendo eShop purchases can be redownloaded at no cost.
Also in the link provided there is a guide on how to delete your game without losing your saved data.
- Select the data you would like to delete, then select "Delete."
- Tap "Delete Software and Save Data" or "Create Save-Data Backup and Delete Software."
- If you have selected to back up the save data, you can do so using an existing folder or you can create a new one.
- Tap "Delete" again to confirm.
And here is a guide on how to redownload games from the eShop.