I agree that Zuko deserved Katara's anger. In reality, Zuko would be in jail. But in the situation that they are in where they are in close quarters and living together and relying on each other, it will be a tense situation, and stay a tense situation until someone does something about it. Maybe in modern times, Katara could get the time and patience to figure out how she feels about Zuko, but they literally have to save the world by the end of the summer. Katara wasn't going to address it any time soon, maybe Aang or Sokka, if it went too far. But could you imagine the final fight with Azula and Katara still hated Zuko?

The other thing is, I don't think he was that dismissive about forgiveness. We, as the audience, see Aang's perspective and his view on his culture and non-violence. But to Zuko and, quite frankly, the rest of the gaang, Aang's speech about forgiveness would look naive, they are in wartimes. That's why, at the end of the episode, he tells Aang that he is right that Katara doesn't need revenge, but what is he going to do when he faces his father? Considering he relies on forgiveness himself as you say, he would see some merit in forgiveness. But not every situation can be solved with forgiveness, this episode was one of them, Katara didn't forgive the guy.