***Potential spoilers below*** (not sure if any of this merits spoiler tags)
Not the Joker because of the age gap with Bruce? Or because of the personality?
As far as personality, we have no idea what kind of Joker this character can be because this film was about the Arthur Fleck persona for almost the whole story. But Arthur Fleck no longer exists at the end. The Joker beyond this point could easily be more of the evil genius psychopath that he's traditionally associated as being.
Arthur Fleck was timid, reactionary, and had virtually no charisma whatsoever. But that all changed. Arthur had been living a lie about who he was and where he came from. He'd also been heavily medicated to suppress/alter his cognitive functionality. When he discovered his true background, and then went off his meds, he became a whole new person.
In that awesome apartment scene with Randall and Gary, Arthur used an Australian accent with Gary. He then spooked him by grabbing his leg just for a laugh. All that (even the way he barely held the door open for a while) was a new level of charisma. And grabbing the scissors, locking the door, playing a role . . . all foresight, strategy, and confidence. No longer just reactionary and timid.
If they did a sequel (and I hope they don't) set 10-15 years forward, the sky's the limit for what the Joker persona would actually become. His mind, free of the self-delusion to mask his pain - and free of meds, might actually be brilliantly capable of Joker-type mayhem. I could totally believe that if they were to ever go there with JP's Joker. Maybe that's just me, though.