What would... the Jester and Gorosie!y/n's relationship/interaction be like?
I feel like Gorosei!y/n wouldn't really take any of her deaths seriously considering she can switch her consciousness into her clone.
Aside from dissecting and performing horrible experiments on people, they have other things they like as well. With that being said I hc y/n like plays. Jester's ability to clone would come in handy for playing multiple roles at once.
(if the story requires a character or more getting killed, she'd actually have to kill her clone to fulfill y/n's standards. Unfortunately).
That and some good old bantering to see who can outsmart the other.
It's pretty harmless as long as she doesn't quip anything that's a tad too relatable to y/n's past (regardless of whether it's intentional or not)
The Jester would perform similarly to how she does with Saturn, which does indeed mean theatrical plays. However, her public performances are completely up to her, albeit with some minor input from her superiors, so I believe that Y/N would primarily let her do her own thing to see how it turns out. Otherwise, I think Y/N would make the Jester perform plays based off of recent scenarios (dare I say something like Marineford), or based off of books. Sometimes, the Jester doesn't even know what the play is about, and has to guess her way along while trying not to be shot and killed.
I do also think that Y/N despises when the Jester says anything about their line of work or how they do things in the W.H organization, and is the most likely to kill her because of that. To them, she is entertainment, not something to be opinionated. The occasional banter is nice, it's amusing to think that the Jester knows anything she's talking about, but then she says something a little too close to home or insults her way of controlling her subordinates, and suddenly her own corpse is laying infront of her.