Along with the premiere date,
it seems that the next crossover is going to be between Superman & Lois and Batwoman, and this is where they're introducing Bruce, right? What do Lois/Clark and Kate have in common beyond that they all know that guy? In which case, two options:
1. BRING CONROY BACK AGAIN, YOU COWARDS. And this time put him in a Batman costume and weather the legal consequences.
2. If not that, Arrowverse Batman must be a knock-off Green Arrow just as its Green Arrow was knock-off Batman. Van Dyke, awful chili, yelling about fat cats. Literally have the reason he vanished be that he divested himself of his assets because he realized it's impossible to be an ethical billionaire.
(In principle I'd also dig the idea that Batman vanished because he's down in the Dark Multiverse being tortured by Barbatos with Batwoman and Superman mounting a rescue mission, but realistically when these shows inevitably do their version of Metal it ain't gonna be a two-parter, hilarious as it would be to finally introduce Batman by bringing in ALL THE BATMEN.)