The next topic I've been discussing with my interlocutor is the topic of Mary. He naturally objected to the title of Mary as Mediatrix, and was concerned that Catholic devotion sometimes substituted Mary for God. (He had a plausible reason to believe this, citing some over-the-top devotional literature it was my duty to do damage control for.)
Evangelicals and Fundamentalists object to the title of Mary as Mediatrix (the feminine form of "mediator") because of 1 Timothy 2:5, "For there is one God and one mediator between God and men, the man Christ Jesus". Ergo, there are no other mediators, Catholicism is heretical, case closed.
I pointed out that Mary's role in Catholicism is to bring us to Jesus. Note that it is not to bring us to the Father; that is Jesus's role only. This verse means that there is one mediator between God the Father and man. It doesn't preclude there being mediators between Jesus and men.
I invited him to consider the role of the evangelist. Is the role of the evangelist not to reconcile men to Jesus? Isn't he trying to bring them to Jesus? Doesn't he pray for them and intercede for them and answer their objections and explain how much God loves them? That's mediation. It doesn't compete with Jesus's unique mediation because the evangelist mediates with Jesus, and he mediates with the Father. So it is with Mary. Her role, like the evangelist's, is to bring people to Jesus. That's it. That's what "mediatrix" means.
We do have to be careful in Catholicism to not exalt Mary above Jesus or God the Father. We have to realize that her role is to bring us to Jesus. It certainly doesn't mean that we shouldn't pray the Rosary, or be devoted to Mary, but we must never think of her as a destination or as a substitute for God or Jesus. "To Jesus through Mary", as they say.
I presume that you as a good apologist you cited examples of mediators between God and man in the Old Testament (especially Moses) and examples of Mary's mediation with Jesus in the Gospels (especially the miracle at Cana).
Also I see from the third paragraph of your post that you explicated well the two famous Latin statements relating to that topic: "Ad Jesum per Mariam" (To Jesus through Mary) and "Per Christum ad Patrem" (Through Christ to the Father).
Did you mention Mary's role as a mediatrix between God and man by bringing God to man through her virginal conception and birth of Jesus (the Incarnation)?