So in continuation of this meta, @nicolovesmonstrosity mentioned how Jerome might see Jeremiah in Bruce as well and oh Gotham how you haunt me like the aftertaste of a fucked favourite childhood desert.
See, to me Jerome plays with Bruce like a Jeremiah-shaped ragdoll he just stole from an expensive toy shop. The first time they meet each other, he takes Bruce hostage with a knife to Bruce's throat; in Jeremiah's introduction to the show the first time we see him on screen with Jerome, he takes Jeremiah hostage with a knife to his throat. In fact, Jeremiah's introduction wholly revolves around whether he lied about being held at knife point by Jerome or not.
Jerome also does clown makeup on Bruce in the possessed circus fiasco, and Bruce looks like the world's saddest clown. Later he's tied to a post with a massive cannon, a phallic shape, that Jerome points at him. In S4 Jerome posthumously puts a permanent makeup on Jeremiah's face with the insanity gas and Jeremiah looks like the saddest clown ever. Miah keeps pointing a gun, also a phallic shape, at Bruce in different scenes.
The way Jerome plays with Bruce is the way he continues to play with Jeremiah, even when Miah is not there. To me Jerome becomes obsessed with hurting Bruce as a part of his games because he sees in Bruce the sort of intellectual untouchable cunning characteristics that Miah possessed. Bruce looks innocent and childlike and he is anything but,, he trembles and shivers but he is also courageous and smart and always finds a way out of Jerome's story, the way Jeremiah did too. Bruce also is arrogant, high and mighty, obsessed with being the better one, the good one and pre-insanity gas Jeremiah thought he is the good one too. Jerome tells Bruce in the Hall Of Mirrors that they're alike, he also tells Jeremiah that he's just as dirty and bitter as him. In Bruce Jerome wants to destroy the kind of innocence that he continueed to destroy in Jeremiah, and there's also the talk of how Jerome trashes Wayne Manor first thing after he runs away from GCPD. Jeremiah escaped into a lavish high society lifestyle with wealthy parents after he left the circus, clawing at Wayne Manor is how Jerome deals with not being able to be a part of Jeremiah's new extravagant life and not being able to destroy it and drag Miah back to hell with him.
So like, Jerome is absolutely ill-enamored with Bruce because Bruce reminds him of a well off teenager Jeremiah that Jerome can enact all his violent fantasies on.