That's just it, though -- I don't think she was being fake. I think she was genuinely trying to put Kate at ease and convince her they were on the same side. She was led to believe that Clint killed her sister Natasha, and she saw that this promising girl with whom she has more than a little in common had fallen under Clint's spell, and she was trying to free her from that, much like she was working at the start of the episode to free other Black Widows from their brainwashing. (Seeing the movie would have helped give you more context there. It's just a couple of clicks away on Disney+.)