Bats is out of his league. Cap is a highly trained solider, tactician, strategist, and has killed more Nazi's and Communists then Indiana Jones, Hellboy, and James Bond combined. He also heals fast and never gets tired. Bats is a detective who has a lot of idiot fans who scream like the children they are if he isn't perfect at everything. Anyone who thinks Bats would have a chance in Hell, doesn't know shit about Steve Rogers. He'd kick Batman's ass then force him to do 100 push-ups.
JLA has some uber-powerful members. God like really. But so do the Avengers. Maybe not quite as powerful, but close. But the Avengers are more flawed. More human. More real. Their human flaws expose them to other ways of thinking, as opposed to the always prim and proper and there for inhuman DC characters. The JLA could easily overpower them. But in a prolonged fight the advantage goes to the Avengers and their ability to adapt like all good soldiers. Because Avengers don't always have the hang ups about killing if they have to. And for any Batman fan screaming about how "Batman could create an anti-Avengers gun to turn them all into bunnies!". Well I got three reasons why that won't happen: Tony Stark, Hank Pym, and Bruce Banner.
