Batman admitted that when he saw the Bat-Lamp light up in midair, his mind had crashed for a moment.
In the past, to keep people from discovering his true identity, he had deliberately arranged for Bruce and Batman to appear at the same time, creating alibis for himself. But this was not his universe. This universe had never had a Batman, and this universe's Gordon was already dead. So who had lit the Bat-Lamp?
Batman gave it some thought. He was certain that an outsider was behind this, because the Bat-Lamp's design was familiar. Though its bat emblem was slightly different from his, it could not have been invented by locals in a universe without Batman. It could only have been made by someone who had seen a Bat-Lamp before.
Quite a few people in this universe had seen a Bat-Lamp. Excluding himself, and excluding Joker-man from Owlman's universe, that left Main Universe Joker, The Batman Who Laughs, and Schiller.
Who had lit the Bat-Lamp?
Any of the three was possible. To achieve a certain goal, none of them would mind using a method like this. But Batman had a strange intuition that The Batman Who Laughs might have done it.
If he had to give that intuition a reason, it was this: adding elements of an orthodox Batman to this universe would seem incredibly boring to the other two.
Though Joker was always interested in Batman, what he liked was exploring the element of Batman in depth, not treating Batman as a jigsaw piece to assemble into something else. He would consider that blasphemy.
For example, if Batman were a plant, Joker would be a botanist. He would rather explore what kind of soil could grow that plant, what could help it flourish, what its fibrous veins and structure