I don´t think it´s that Natsuki is malnourished though. In the first act, if you decide to help Natsuki during the festival. She states that Sunday that when you meet up with her that her dad MAKES her eat all of her food. But then again, she may be lying and refused to take the cupcake because she was scared her dad would find out she ate. I´m not sure. But, I don think she is being starved. I do like to here these conspiracies though!
Actually I think she is malnourished, though. She might eat all of her food because he usually never cooks. In the "Papa" poem, it says she likes when he cooks, which could mean he doesn't do it often. Also, it says she likes when he keeps food in the house. Plus, she is being abused (duh). Just my guess, though. I like your theory, though. Much more positive!