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I feel like even before Hawkmoth knew that Chat Noir was his son, he had less of a grudge against him than Ladybug.
See, Hawkmoth understands the destruction that Noir symbolizes. He's a bitter old man who lives in stillness and isolation.
LB and CN may do the same thing, but she's more of a philosophical threat to Hawkmoth than CN. She is literally the symbol of Life Going On, the outsider to his "family", a cosmic message from the universe that his grief is overdated and poisonous. She reminds him that living in the past is pointless, and if he doesn't wake up and pay attention to all the dangerous, wonderful people existing in the present, they will crush him.
Ladybug is Life, and Life is not sympathetic. It coddles no one, it waits for no one. Just look at the African savanna.