So the problem here is that the character who believes he is the Joker, will likely act in a manner consistent with the character from the DC Comics and parent Waner Bro. This brings the character dangerously close to DC's character to not be protected with a fair use defense. Fair use would protect you if your character was described as being "crazy like the Joker" or some similar phrasology where the character is not immitating the DC character, but is likened to the character as a way to explain this the audience.
But since your character believes he is the Joker and acts like him as part of his insanity, this is much closer to copyright infringment to pass the fair use test.
With that said, it's not out the realm of possibility to use the idea... but rather than say he's imitating a DC Comics Character, make an in universe fictional comic book company that, every reference, makes clear its the DC comics with the serial numbers filed off. This isn't unheard of, especially in the comic book industry, as both Marvel and DC have a group of villains that are suspiciously similar to a heroic team of the competitors. Marvel has The Squadron Supreme so the Avengers can fight the Justice League and the Shi'ar Imperial Guard so the X-Men can fight The Legion of Superheroes (in the Guard's case, the artist who created them had previously drew for LoS comics before going to work on X-men). DC has the lesser used Maximums (with the battle cry "Maximums, March!") that are clearly the avengers (they aren't used nearly as much as the marvel examples).
So if you created a character who is immitating a clown based supervillain of the well known comic book character of Catman and Jay. You can even preserve a playing card theme by naming the fictional character being immitated "Jester," "Wildcard," "Bug," "Jolly (or Jolly CLown)" or "The Fool" of "Le Mat" which are all related to the Joker playing cards (Jester being the typical design of the Joker, Wildcard and Bug refer to the Joker's role in games when it is used, Jolly and Jolly Clown both make reference to the Italian name for the character on the card and "The Fool" is a card from a Tarot Deck which often plays a similar role to the Joker in Tarot decks, though usually not the same way. In trick taking games (War or Hearts as an example), The Fool always loses the hand, but is always a legal play. It's used if you have a high trump but do not want to lose it during the current hand. In Ocuult Tarot, the fool represents great luck or fortune, so it's not a terrible card to have and many heroes heros are given a The Fool Motif. "Le Mat" or "Il Matto" is the name of the fool card in certain variations of Tarot decks and occupies the same role in play. The term is an archaic word for "The Madman" or "The Beggar".