Could you just change the Mii being used by going to Mario Kart Channel, and then Options? You'll be still able to continue with the current progress you've done so far.
If you still want to fully wipe out the saved data from the game, you can simultaneously hold A, B, X, Y, L and R while booting up the game until it prompts you to delete everything from the game's saved data.
Resetting your 3DS will definitely unlink your NNID account, which will retain your eShop purchases while losing your saved datas for every game in the system but the game card ones.