The point is, Lucy is so, so, so Cooper-coded, and the more we learn about him, the more we can see why Lucy’s attitude irritates him so much. Basically, the guy has been beating himself up for 200 years for being an idiot, for trusting the Establishment, for being stubborn, gullible, and naive, precisely the way Lucy is in some ways. So seeing her is like seeing himself from so long ago, hoping in a social structure that’s nothing more than smoke and mirrors, with moral values that, in reality, never actually apply. And his problem with her isn’t really with her, nor with the fact that she’s a VaulTec creation (though it might seem that way at first) it’s a problem with himself, with accepting his past self that he’s tried to kill off completely, but that always comes back to him somehow.
What I think both Cooper and Hank don’t get is that Lucy is not how they think she is at all. Yes, there are a lot of parallels between them, but she’s not Cooper. And second, this is something I think is made pretty clear in the last episode: Lucy is not a kid. Her father treats her like a dumb little girl, and Cooper treats her like a naive young girl and she's not any of those things. What it's ironically amazing about this is that then there’s Norman, who knows exactly what his sister is capable of and doesn’t doubt for a single second that Lucy could have survived the Wasteland. He even says, “You don’t know my sister,” because in a way he’s saying that he grew up with her and knows she’s the most stubborn, headstrong, annoying person in the universe and if she sets her mind on surviving the Wasteland and seven nuclear wars, she will do it, because that’s who she is.
And in a moment of desperation, before he even tries to talk to his father over the radio, the first person Norm turns to is Lucy, because Lucy is his fucking older sister, and he doesn’t see her as an innocent girl or daddy’s little girl. He sees her as a reference point, as someone who raised him in the absence of a mother, and as someone totally capable, not only of surviving every single danger in the Wasteland, but of coming to find him and saving his life. Everyone underestimates Lucy Maclean except her fucking little brother, because he doesn’t see her as a kid, he sees her as a role model.