Katara definitely doesn't perfectly fit the "Ice Queen" trope because very caring and isn't as ruthless as characters such as Azula, although some may argue that she was certainly cold towards Zuko. However, Zuko does spend some screen time trying to defrost Katara and regain her trust because I think he knew that in order to get closer to Katara, he had to prove he wouldn't throw her trust back in her face, like he did the last time.

Also, I don't think Katara really hated hated Zuko since she saved him from falling to his death in TSR. She just needed to know that he was very sorry for what he did and that she could trust him.

Some may argue that the pairing doesn't fit the trope because they're not a canon couple but subtextually I think it works.