Let’s make this scenario more interesting.
Yellow Circle Batman absent-mindedly dials the Batcave – and he himself answers the phone, away in the cave. Each instantly recognizes the voice on the other end of the phone as being his – uniquely him.
“Hello?”
“Who is this?”
A pause.
" I want to know who you are."
“You know.”
They both hang up. Unpleasant possibilities occur to them both. After a few minutes, Yellow Circle Batman calls the Batcave again.
“It’s me.”
“Yes. The question is: what are you?”
“Don’t.” Batman warns. “This is stupid.”
“No. It’s anything but.”
“I’m not the one who’s deluded here. You’re the duplicate – a Martian shapeshifter, a doppleganger, some dimensional replicant, an abortion of Hypertime.”
“Or YOU could be the fraud, here. Some metaphysical fluke given corporeal flesh and surreal bone. Or a nightmare. Or a virtual simulation conjured by one of my enemies to torture me.”
“I won’t surrender.”
“I won’t break.”
“Then…?”
“Yes.”
“You won’t just …disappear…?”
“Not until everything I own is mine again.”
They both hang up.
Yellow Circle Batman (YCB) is in the middle of downtown Gotham: immediate assets: he has with him the Batmobile, some computer equipment in the trunk, his belt, a change of clothes for Bruce Wayne, Bruce Wayne’s wallet, a few hundred in cash. Yet he has the freedom of the whole wide world around him. Yet due to the intensely interpersonal nature of the fight, he will not enter the Wayne Manor Estates.
Plain Logo Batman (PLB) is hunkered down in the Batcave; his immediate assets: he has the entire resources of Wayne Manor and the Batcave at his disposal, including his other Bat-vehicles, his files for his criminal cases, explosives, poisons, etc. Due to the intensely interpersonal nature of the fight, he will not leave the Wayne Manor Estates.
Up for grabs: WayneTech’s resources. Bruce Wayne’s banking accounts. The Wayne reputation and prestige. Allies, like James Gordon, Bullock, Robin, The Huntress, Nightwing, Azrael, Catwoman, Lucius Fox.
Out of bounds: Anything OUTSIDE traditional Batman mythos, or in fact, outside the Greater Gotham Area. That goes double for the JLA and their equipment. No Superman.
Nuetral: Oracle… she doesn’t know who to believe, so she helps them both, when asked, EXACTLY as ordered. Likewise, Alfred Pennyworth.
Yes, I swiped this plot from Harlan Ellison. Knock yourselves out.