Hawkeye wasn't quite a founding member of the Avengers, but he was a part of the second wave of formal recruits who joined Earth's Mightiest Heroes shortly after the group was founded.
Clint Barton is a great character who brings so much to the table with his prolific aim, his impressive hand-to-hand combat and the sheer imagination he has in turning everyday objects into deadly weapons. But for Barton, the sad truth of the matter is that Batman would find him the easiest Avenger to take down.
The Caped Crusader is frankly better than Hawkeye on too many levels for Clint to stand a chance against him. We've seen Batman battle and beat Green Arrow in the pages of DC Comics at times over the years, and any such fight with Hawkeye would go the same way.
Without trying to dismiss Hawkeye as a total loser, Bruce Wayne is simply smarter, stronger, more bullish, has a wider array of gadgets available to him, and is a better stand-up, straight-up fighter than Hawkeye.
In terms of the reality of it, Bats would look to get close to Barton, minimalizing the chance for Hawkeye to use his eagle-eye to pick the Dark Knight off. The excessive ninja training of Bruce means that he's able to perceive shots at a slower pace than your average superhero, so that's in his favour when faced with Hawkeye's attacks - and once he gets in close, it's game over for Clint.