Why would she start caring about the kid now? She has a 3yo kid yet found time for infidelity. At best giving the kid some thought every now and then strokes her narcissism. Even if she manages to keep her husband and maintain a husk of a marriage it's not going to be great for the kid to grow up witnessing it. She's in love with the idea of being a woman who can wrap a younger man around her finger despite being married with children, or perhaps with the drama of it. Her love for OP isn't necessarily "fake" but it comes from a sick place. This may not be hitting on the truth %100, but there is plenty of cause to say she is amoral, manipulative, just overall ugly on the inside.

If OP wants to keep her he needs to play into her uglier side. Yours are good suggestions, but it's only about the kid to her because one of the questions in her mind is "How will I look if I do this?" He needs to be something she wants to show off to the world. She might also enjoy getting a rise out of men etc., and if that's the case the husband needs to be outdone.

Of course I think OP should be happier with any other woman, since they would actually deserve his love and care for him in a healthier way.