It didn't really bother me because they took pains to show that he wasn't particular nibble or athletic. They showed him getting hit as much as he gave, almost being overwhelmed by thugs, which in turn gave me the impression he's wearing excessive amounts of armor which makes it difficult to move. You see every shot bouncing off his armor. I thought in a way it made it more realistic than Nolans batman who becomes a martial arts expert and ninja long before hes batman. It would bother me more if they portrayed him as invincible or superhuman, but to me he just looked like he was walking armor in an armored suit similar the villain in Rampage (2009). I got the impression hes covered from head to toe in Kevlar and plates which would explain why he stumbled around quite a bit in his fights.Only read it once over a decade ago, but I still remember the tension in Year One, where Batman needed to do his best to avoid bullets and barely came out of that church alive. Likewise, that scene where he's hunted by the police in Mask of the Phantasm. They can think of reasons for him only wearing textiles, like stealth, speed and aerodynamics. Or set it in the past, when machine guns and semi-automatic pistols weren't so easily acquired yet. In the movie The Friends of Eddie Coyle, selling machine guns illegally gets you life in prison. Similar laws in Gotham would make the bad guys want to carry fewer machine guns. They can put a yellow bullet-proof bat symbol on his chest to draw the enemy's aim. But when it gets to the point where Batman takes bullet after bullet, shotgun pellets at point blank range and even (what looked like) high-powered sniper rounds, that's boring.
I think when it comes to suspension of disbelief it's all in the eye of the beholder. They gave me enough excuse to ignore his near invulnerability. I think the only scene that made me blink was when he injects himself with adrenaline. Thats not exactly superpower inducing so it annoyed me when he just goes about his business. In reality I think it would give him a brief second wind, but probably a heart attack if he kept going like he did. I'm not the biggest fan of that kind of mcguffin, but I got over it.