I don't see Hamlet in the movie. Phil does not want to steal Rose. He wants to destroy her. This is what drives Peter to take action. Remember that Peter had to cut his father down from a noose (the father committed suicide). It is clear from his conversation with Phil that the father was an alcoholic and obviously depressed. Peter fears the same fate for his mother. If you recall, he refers in the film (toward the end) to the necessity for a son to save his mother. Peter has to save his mother from Phil.

Hamlet is about inaction. For almost the whole play Hamlet worries--what to do? He soliloquizes. What to do? He spends the play struggling with his conflict. Peter doesn't struggle. He is clear in his purpose. Just as he was able to dispatch a rabbit quickly and bloodlessly earlier in the film, he is able to deal with Phil. Peter is quite clinical in his actions. He needed a rabbit to practice surgery, so he killed it. He needed once again to put a rabbit out of its misery, so he killed it. He doesn't stop to ponder whether he should or shouldn't. He doesn't bemoan the fact that he has killed a rabbit. He's very practical and proactive. Hamlet is the opposite. He is frozen by conflict.

Thank you for that interesting response, @goldgrifin007. I'm glad you watched the movie. It's good to get a new perspective.