When you start up Mario Kart, it will try to use the name already in your DS, but it will ask for you to approve or input a new name specific to Mario Kart that doesn't change your DS settings.
Your My Nintendo account name is never seen by anyone. Your My Nintendo screen name only shows up on their message boards and such.
Interestingly enough, while your Mario Kart name shows up to others when you play, your DS name is what will show to others browsing the WFC site from the PC.