Batman can't move faster than a bullet, so protection is more necessary than dexterity. What's the point of being able to move fast when one bullet can easily cripple you? That's all it takes. One lucky shot to the shoulder, leg, arm, etc. No matter where it hits, it's gonna slow him down.
A cloth suit is absolutely horrible. Just look at Dead End. It's a ****ing joke. The current suit is the best we'll probably ever get in a movie for many, many years. In the real world, Batman would need the protection he has. He's not Spider-Man or Superman where he can easily dodge them.