Or am I confusing the incarnation and the trinitarian generation to an intolerable level again?
I think so. The Incarnation comes after the Trinity. The relation between the Father and Son is generation, or paternity and filiation. The relation between the Father and Son, and the Holy Spirit is procession or spiration.
The More Than Three Persons discourse does not answer the co-paternity of the Holy Ghost quandary directly. Reply to Objection 4 says that since there is a relation of proceeding between Father and the Holy Ghost, the Holy Ghost cannot be in relation of principle even as parent. But Aquinas did not apply the same logic when the Son, though in relation of begetting with respect to the Father, is in relation of principle as spirer.
Moreover, if we know that the Father and the Holy Ghost have one and the same goodness, then it would follow that the fruit of the goodness, the Son, would be co-begotten by them.
The reply to Objection 4 would still stand, because of the fact that the infinite loop purported in Objection 4 would have to alternate between procession and filiation and therefore break, leaving the number of Divine Persons at 3.