First, if there are any difficulties with my explanation of the "equal papi" rule, please tell me. Karnoffel is only a grandparent, I want to emphasize; a different game, with permanent trumps (as I imagine for "VIII Imperadori"), is the parent. But what I've just said is not the explanation, which I've already given and see no need to repeat, although I will if anyone needs it.
About your explanation of the "equal papi" rule, Nathaniel, I see a few problems. First, everywhere else seems to have managed to come up with hierarchies they could agree on, at least at that time and place. It isn't that hard. What is different about Bologna? The only thing I can think of is its form of government: signorial rather than by a hereditary family. In that way it is like Florence. A hereditary ruler who enjoys card games could simply dictate an order. The signoria might not have wanted to bother spending meeting time discussing it. But still, you'd think that some specific order would dominate after a while, especially once the Bentivoglio were the leading family - unless, of course, they were the ones that wanted the four to be equal, so as to undermine the idea of loyalty to the papacy alone. That in itself is an argument against the rule being post-Bentivoglio, because after them the papacy ruled. And despite Phaeded's four hats, two of the hats on the cards simply aren't papal. (Please be clear that I am not addressing the issue of gender at this point.) Having an emperor trounce a pope isn't how the papacy likes to see things.
Another problem is that if the pope is agreed as highest, then it shouldn't be one of the equal papi. There should be just three fighting it out.
Added an hour later: Actually, I may have answered my own question: what makes Bologna different is precisely that it was struggling, with the aid of the Bentivoglio, to get out from under the papacy. Having an emperor trounce a pope is pretty nice. But I don't think it is enough in itself, because there would have been plenty of people who thought it was the Ghibellines up to their old tricks. There has to be the precedent of an earlier game with equal trans-natipnals as trumps. Added later: Also, Piedmont still has to be explained.
Now about gender: while your explanation of Florentine Minchiate makes perfect sense, I don't see that it applies to the Bolognese case. In Bologna there were two types of kings, old and young, and it makes sense that both types would be represented in the Emperors. The corresponding differentiation among queens was profile vs. full face. But it didn't suit the papals' portrayals to have them in profile, because then the one in profile would be seen as looking at, and so subservient to, the other one. That's ok with queens, because they're a lower rank, but not with papals. Here's a head-to-head comparison, using the full-faced queens:
There are 2 reasons for using queens as models for papals: (1) they are the closest in rank to kings; and (2) popes were clean shaven then. Mind you, I'm not saying that they were all-male from the start. I just can't see my way clear one way or the other.
Another thing is about the "papotes", as they were called. To me this signals that some people from Piedmont recognized the affinity of their game with that of Bologna. I would expect them to have been students rather than courtiers, because there wasn't much court-to-court interaction between the two that bypassed Lombardy, where the term would not likely have been used. I can't see that the term itself says anything about when it was introduced.
I suppose the rule itself could have been introduced by aristocratic students coming home from Bologna, before the game as introduced from Milan was established, but I like the idea of it coming from Lombardy instead, more authoritative. Both it and Angel-high are perfectly compatible with the 15th century Milanese cards.
One final thing: the "papi" and "mori" in Bologna have never been numbered, except by a few players hand-writing numbers on them. The numbers on the BnF copies, as I expect you realize, are on all the triumphs, in Marseille order. The numbers, when they were added, started with 5 Love and ended with 16 Stella.