Is anything incumbent upon a pilgrim performing Tamattu' Hajj if he has intercourse with his menstruating wife after the first stage of exiting the state of Ihram (Al-Tahlul Al-Asghar) and before Tawaf Al-Ifadah?

What you did, by engaging in direct physical contact with your wife after completing all your Hajj rituals, does not invalidate your Hajj nor does it require a compensatory sacrifice (dam). As for your wife, she owes a compensatory sacrifice (dam) because of your kiss, as she was still in the state of ihram for Hajj. What is incumbent upon her is to sacrifice a hady (the minimum of which is a sheep), to be distributed among the poor of the Sacred Precinct of Mecca. Her Hajj is valid, Allah willing.