Stark understands that a billionaire training himself to the peak of physical perfection is like a billionaire learning to grow his own organic tomatoes. The whole point of money and technology is not having to do stuff with your bare hands. And for every "Batman could design something to disable the armor," I'd raise you an "Iron Man's actual job is designing things and he knows more about the armor." Sure, if they were stripped naked and locked in a room, Bruce would utterly dominate Tony, but if that's what you want to see, there are websites you can go to.
OK, Tony Stark was an arms dealer. That's a challenging origin for a good guy. But his father built the company when weapons merchants were heroes, because Hitler. Howard Stark reversed the polarity of Godwin's law. His entire legacy was "No More Nazis" and "Yes More Captain America," which means he couldn't have been a better advertisement for weapons technology if he were a zombie slowly shambling toward your daughter. Tony genuinely believed that giving the good guys guns was how you fixed things and sacrificed billions to stop the very second he found out he was wrong. The very instant Bruce Wayne didn't get what he wanted, he stormed off to a remote prison hellhole, and while I'll admit that's the most badass version of screaming "I'm going to my room!" and slamming the door possible, that's still what it was.
And when most rich dilettantes run off to a sweaty house full of mid-Asians to wrestle with, they have a much better time.