Purse is the correct answer.

FYI, aside from meatball, all of them are in the anime.
Meatball is probably a mistranslated of dango, which is in the anime

Naruto eats ramen, that's the whole plot of naruto
Naruto have a purse, from which jiraya stole money
Anko eats dango
Choji eats BBQ, usually with his team

As for game:
Ramen is an exp boost item
There are purses in plot instance
Dango/meatball is something we have too much of
BBQ is an exp boost item.

So, all 4 exist in BOTH the anime AND the game. The ONLY way the question can work w/o changing the answer is:
Which of the following does not exist as an ITEM in the game.