In your headcanon why was Katniss offended when Peeta claimed that she has gone back and forth thinking of killing him, in light of her claim that killing Peeta would be like putting down a capitol mutt?
She's offended because it's more complicated than that. Katniss doesn't actually want to kill Peeta. And she tells us that. She tells Peeta that explicitly and truthfully. Other than the short time frame where she thinks Peeta is helping the Careers hunt her down and kill her in the 74th Games, Katniss never actually wants to kill Peeta, not even after he's hijacked. Her comment that killing him would be like putting down a Capitol mutt is very much an emotional response to the ways she feels hurt and embarrassed and quite frankly, devastated by what Peeta has become. It's not a real declaration of intent to kill him.
Meanwhile, Peeta's claim that Katniss has gone back and forth thinking of killing him is likely derived from a combination of the things the Capitol showed him/led him to believe during the hijacking, and Peeta's developing hindsight views of their relationship and interactions as he recovers from the hijacking. The way he phrases it implies, not that she actively wanted to murder him, but that killing him was always an option lurking in the back of her head. Which... it kind of was constantly there, given the fact that they kept getting thrown into televised death matches where only one of them could survive... it's not that far off, is it? HOWEVER...
After about an hour, Peeta speaks up. “These last couple of years must have been exhausting for you. Trying to decide whether to kill me or not. Back and forth. Back and forth.”
That seems grossly unfair, and my first impulse is to say something cutting. But I revisit my conversation with Haymitch and try to take the first tentative step in Peeta’s direction. “I never wanted to kill you. Except when I thought you were helping the Careers kill me. After that, I always thought of you as . . . an ally.” That’s a good safe word. Empty of any emotional obligation, but nonthreatening.
“Ally.” Peeta says the word slowly, tasting it. “Friend. Lover. Victor. Enemy. Fiancée. Target. Mutt. Neighbor. Hunter. Tribute. Ally. I’ll add it to the list of words I use to try to figure you out.” He weaves the rope in and out of his fingers. “The problem is, I can’t tell what’s real anymore, and what’s made up.”
-- Collins, Suzanne. Mockingjay chapter
Peeta frames Katniss killing him as a choice that she would have wrestled with, which in turn indicates that he's come to view his relationship with Katniss in terms of when he was expendable to her and when he was either necessary for her survival or simply a nuisance but not worth killing.
But to answer your question, she's offended because while Peeta is correct in his assessment that the choice of whether or not to kill Peeta has always haunted Katniss's thoughts since the moment he was reaped, his comment is a grossly unfair summary of that situation. It is never as simple as deciding whether or not to kill him because their lives have become so entwined both by Capitol design and through their own individual choices and desires. His list of words illustrates just how complicated their lives and thus their relationship are. She's offended because Peeta is oversimplifying Katniss's feelings and her struggles with juggling her own wants with her survival and with her humanity. It is actually a grossly unfair statement, so I think she's fully justified in being upset with it.
That being said, here she at last acknowledges that Haymitch was right that she should be helping Peeta, not constantly getting angry and frustrated with him. And also something that she outright states later on, and she also acts on it: If she wants Peeta back, she has to help him fight his way back, because so much of his source of confusion is going to center squarely on her and their relationship.
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