Maybe later! I'm about fifteen minutes away from crashing right now, and long tl;dr is long. Bullet points are:

-She should have apologized. The goal was to teach Aang waterbending. Aang is the Avatar. His learning it is more important than her. She knows this.
-She didn't care. Waterbending had been her dream since forever, it was who she is, she wanted to learn how to fight and how dare he stand in her way for something so stupid.
-She knew that Pakku would beat her. She didn't care. She was determined to do everything that she could to prove that he had to take her seriously as a fighter.
-She almost did apologize. But something in her just snapped at the injustice of it all. And she doesn't look back on challenging him with anything other than a sense of rightness. He was wrong, she was right, and she's just glad that he was finally able to see what his stupid commitment to male superiority (or 'his culture) had cost him.
-At the core of who she is, Katara is a fighter