I like how John Byrne did it in the Captain America/Batman one shot. They ultimately teamed up (of course) but before they realized who the other person was Bruce Wayne briefly engaged in a fight with Steve Rogers. Neither had costume or accessories (batarangs, shield, etc.) As they fought Rogers just barely started to get the upper hand until Wayne figured out that he was really Captain America just seconds before Rogers figured out that Wayne was Batman. Once Wayne pointed that out they ended the fight and shook hands.
So Byrne painted Rogers as the slightly better fighter with Wayne slightly edging him out in deductive reasoning. Pretty cool and respectful toward the strengths of each character.