This is incredibly simple. All you have to do is beat the mission and get the food. Then, after you're back on the map, open the Home menu and close the game. When you go back in, the food is available again.

Duuuuuuude. Can confirm this works. I can't believe we all missed something so simple xD Thanks for sharing, TC!

Nice to know when I do start farming for food. I wonder though... if a game restart works, would it also just work to back out into the menu or title screen? Although I guess at least on an N3DS, it wouldn't matter that much in terms of how much time you save. But it would be a bit more on a regular 3DS.