To me, there are so many variables even in "normal" cases (people bet on boxing matches for a reason) that trying to analyze fictional characters from different worlds in different contexts is a quick way to go way off the deep end.
Classic DC, they'd never actually fight. Classic Marvel, the fight would be interrupted by the villains. Some writers would have Batman cheat. Some would have Cap super-strong and a better fighter.
That's not to dismiss the question or the fun, but...y'know, even if they were actual people who could be measured, it'd be a crapshoot.
The ideal version plays out as follows.
Cap handily beats Batman, and is about to kill him with the shield, when Batman realizes that his opponent is actually a Skrull! So he takes the shield and bludgeons the alien to death (modern comics...), then lights the corpse on fire, just in case it's a Martian.
He releases Cap from Luthor's deathtrap (without questioning why Luthor is even in this story), but Cap spots a feather peeking out from under the cowl and reveals that Batman is actually...Owl-Man (pick your favorite version), and the two fight until Owl-Man, about to be defeated, vanishes into some interdimensional villain conference run by Loki or the Ultra-Humanite--like CES or something, but in an even weirder location than Vegas. After visiting several alternate Earths with different-era Batmen, Cap finally stumbles across the Millerian antisocial thug, and the two spar just to preserve the form before the writer realizes that the deadline is approaching.
They race back to New York, where a portal above the Statue of Liberty is squeezing shut. Not sure which Earth is which anymore, but sick of listening to his idiotic brooding narration boxes, Cap flings the Caped Crusader at Lady Liberty's torch and (coincidentally) home.
But is it really Captain America, or is it...a Space Phantom!? Aaaaaaah! Ahem. I said, "aaaaaaaah." No?
Oh, and Grant Morrison appears in-panel for, like, half a page somewhere in the middle, wearing a funny hat from a 1950s Batman story nobody has actually ever read and an Escher print t-shirt, talking about metaspaces and recursive storytelling and pretending he's never read Douglas Hofstader.
And THAT is why I don't write comics. Basically, no matter who wins, the readers always lose!