In what specific literary piece can we apply the views of Plato as stated in Plato's Republic?
This is fairly broad in scope. I wonder if there could be a more limiting function to the question in order to fully grasp where it is heading. If the intent is to find a literary piece where the views of "The Republic" could be applied to find some contrast or agreement, I think that there are many options. It would be interesting for you (or others through enotes) comment on how Plato's Republic compares to Allan Bloom's work, "The Closing of the American Mind." There can be some interesting parallels between Plato's Philosopher King and the role of philosophy and what Bloom sees as the value of the "Great Books of Western Civilization." It would also be interesting to trace how both thinkers construct a paradigm of dire proportions for not adhering to their words. Plato's fear of the poets could be compared to Bloom's view of the modern university that has succumbed to the belief that the Great Books of Western Civilization represent "dead white men" as opposed to works that appeal to the highest of callings.
Not really a good idea. Plato himself discusses Homer frequently, and much of neoplatonic criticism focusses on the relationship of Plato to Homer, and thus a less anachronistic approach would be to consider the Homeric epics in light of not only Republic but Ion as well. Think about whether there are multiple levels of reading implied in Republic, and whether the poet can imitate the forms as well as the sensibilia.
Bloom`s work, although it has some traction in popular culture, is not respected by scholars in ancient philosophy because it uses Plato in service of rather idiosyncratic 20th century ideology rather than contributing to our understanding of Plato himself.