Tywin may also have been concerned about the possibility of poison in his food, knowing there was an assassin in the castle. Demanding that Arya eat the food could be a way to both check if it's poisoned, and if so, whether she was involved and/or aware of it. There have been many hints in the past few episodes that he's aware that she's not being truthful about her identity, and is probably a noble. I don't think he's aware that she's Arya Stark though, because I suspect Cersei has kept Arya's escape secret from him.
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[Book Spoilers] EP 207 Discussion