But that would happen outside of the club I think. Anything within the club would happen under Kathleens watch, as I don't see her going away until the end (if at all), and she can't resist playing her little games. And I'd argue being involved in the carnal side of the club would even be a hindrance in becoming Augusts heir.
I think that would depend on which path we are on. In the truly dark ones, yes, I could see Kath still in command of PC. But on the paths we side with Hana and August, no matter how light or dark, Kath and Warren will be kicked out of the Club.
As for Hana taking a more "active" role in Club activities and her place as August´s heir, I would say again, this would probably hinge on the path. On the light or lighter paths, with a not or only slight to medium corrupted Hana, yes, this could become a issue. But I think as long as she goes into it willingly, without being pushed/blackmailed by Kath/Warren, August would allow it.
But these are just speculations, here is a reprint of my speculations about the Hana paths
For Hana´s story I estimate 4 major branches with variants depending on how strongly (or not) corrupted she is at the end.
- One is the pure Love route with the possible mending of fences between Hana and August in addition. No to just slight/medium corruption of Hana (if MC and Hana want to test out a couple things, if they like it).
- Next is a Love route with MC and Hana being groomed to be the successors of August and Chuck at PC. There are hints of this if you play the Hana route and talk to August and Chuck often. And it becomes clear that August wants to repair the damage in his relationship with his daughter, he just does not really know how, cue us MCs.

Corruption could range from zero (Hana is shown to be pragmatic and thinking clearly on most things) over some/medium corruption (encouraged curiosity despite not convinced at start) to full corruption, if the MC likes his SO to be a loose cannon in bed, e.g. testing porn acting (too much hints of it not only for Hana but others as well), girl-girl, sharing/swinging or a couple shifts in PC to see how it is. (while the majority does not, there are, most often hobby, prostitutes who like doing it)
- Another path would be the Dark Love one, Hana and the MC are a couple, but the MC likes Hana to be as uninhibited and slutty as she can be. This path naturally would feature major corruption efforts, like the ones above and additional ones like e.g. Hana "testdriving" August.
- The final path would be two-pronged depending on corruption level. In this storyline MC and Hana would be friends (maybe with benefits) and if Hana stays uncorrupted or just a bit, then MC helps her getting free of PC, in the case of a strongly corrupted Hana she is in PC for longer.