That’s not to say Earth-1’s Batman will appear on Batwoman any time soon or appear in “Crisis,” though; Supergirl wisely allowed the Girl of Steel to establish herself as superhero and leading lady before bringing in one of the biggest names in DC Comics history.
In all seriousness, I highly doubt that the Arrow-verse would cast its Earth-1 Batman as anybody who couldn’t reasonably appear and recur as necessary, similarly to how Supergirl handles Earth-38’s Superman, played by Tyler Hoechlin.