Its also worth noting that when the Flash alters history by running backwards in time, referred to as the Flashpoint, its Bruce who dies to Joe Chill. Thomas becomes Batman and exacts revenge, killing Joe, then tells a grief stricken Martha Wayne he misses her smile, only for her to become the Joker herself.

The idea that the Joker could be Bruce's half brother isn't much more strange than the idea that his mother could become such a similar villain. This is both likely to be a delusion because its something Joker wants, in order to justify hating Batman, but at the same time, the alternative, Thomas's cover story, that Arthur was adopted, and supposedly proves that Arthur had no relation to the Waynes, didn't anticipate a loose thread, that Arthur's mental health would end up being so similar to his mother's.

P.s., this is not an endorsement of the way these mental health symptoms are treated in psychiatry, nor did the Joker portray them in a very positive light - moreso this is an analysis of what these sorts of symptoms can do to a person or character, and what makes the uncertainty of those situations so scary.