Disclaimer: All I own is an Appa plushie (and if anyone tries to take it from me, I'm going to go all rabid fangirl on them).
A.N.- Since Avatar hadn't given any hints before that Katara resented her dad for leaving-my explanation is she didn't acknowledge her resentment. This has conscious thought (Katara justifying herself, to herself) and subconscious/unbidden thoughts (in parenthesis, italicized).
Reflections
She isn't angry with her father. Really. She misses him, that's all. It's normal to miss him. Her mother may be gone (forever) but he's supposed to come back (hopefully, some day).
It's not like she can grieve for him (yet). She wears her grief for her mother at her throat, a constant reminder of what was lost. It is palpable and weighs her down. She has nothing but her memories to hold onto her father (even though he's still alive-just not here). His lack of presence is a void (empty) that she can barely acknowledge (even to herself).
Sokka has his boomerang (and training) to remember him by. He has no resentment (we were left behind), only a thirst to prove himself a warrior and a wish to make his father proud. When they (finally) hear solid news of Hakoda's whereabouts while they are in Ba Sing Se, she lets Sokka go. Because he's willing to give up the chance for her, no matter how much he misses their father. And she knows she can't just see him for a week (possibly never again) and then leave. Sokka exuberantly calls her "the nicest sister ever" when she tells him to go. She cooly replies, "You're right, I am," but the thought that she isn't the best daughter ever (she can't face him yet) sneaks in her mind as she says it.
And then everything goes wrong (she couldn't stop it). Then she finds herself facing him anyways. So she tries to put it behind her. On some level, she knows she's being disrespectful. But she can't help it. They were fine without him before. It's Sokka's plan and Aang is part of her family (not his son). She didn't think she'd have to face him again until everything was over. She didn't know if she could face him until it was done (would he be there; would she?). When they (if they) got back to the tribe and put it all behind them (like a bad dream). In a way, she lost her father to the war too (as a cause, not a death).
But he is here, and so is she. She knows they will probably be separated sooner or later (not again, but as soon as possible). So the words spill out of her mouth. She may be talking about Aang, but they both hear the double meaning, because it's (family, leaving her) not supposed to be a theme, but she doesn't know how to stop them. And she wouldn't (couldn't) stop them anyways, because it's more than her family, it's the world on the line. It'll take the Avatar and a Water Tribe Leader and a Waterbending Master and more and more to make it stop so they can all go home.
And it turns out he missed her too. But that doesn't change that they have to part ways (she has to go after Aang) and she won't see him again until the Invasion.
If they can, if they make it, if there is an afterwards.