Two hour drive, or fly across the Gotham waters of course, since Gotham likes to take its inspiration from three different cities, namely new York with its island chains connected by bridges. Makes sense though, they seem to be close to each other now
Would be a good way to build his appearance up, have his henchmen stirring up trouble, Natalie gets involved, discovers sorta what they are, well what they definitely represent as her father had to deal with the real one a lot and told her about him.
Consider them knocked silly, she takes after both Bruce and Kate Kane a bit in her in suit treatment of criminals. Brutal like Bruce, and coldly mocking like Kate in outward appearance, both to create fear in the Bat symbol and make them wary or confused of her telltale smirk which may just make her seem like a psycho sometimes to many criminals heh
Good, then maybe they'll start fearing her. She wants all the criminals to fear Batwoman, know she's around, possibly watching their every move, and know what happens when she arrives. Heh. The Bat renown has to be kept and reinforced once again.
Would be interesting, but Natalie (much like myself) is a lesbian.
Could get into that awkward territory Batman and Joker got into though, where he just steadily got into his head somehow and well, Batman kept letting it slide, ensuing many awkward talks on the way to Arkham, forever doomed in a repeating cycle.