I would have laughed (not in mockery, but rather in amused interest).
I would definitely allow him to wear it as long as the other players didn't strongly object. After all, part of the purpose of role-playing is to explore the idea of being a different person or behaving in a certain way.
He wants to be Batman? Great! Let's explore that! How would people in this medieval setting react to that? Well, they would probably think he was insane, or else laugh at and ridicule him. If the player couldn't do something to show that he should be respected/feared/taken seriously, NPCs would continue to react negatively to his character.
Eventually, the person playing "Batman" would have explored the "reality" of being a grown man dressed in a ridiculous outfit in broad daylight in a medieval village, and would either come to the conclusion that this was not a good idea (matured) and taken off the costume and just played a normal person for that setting, or he would get pissed off and leave the game (not matured/learned anything).
It's like in the movie "dodgeball" where the guy tells the other guy "You're NOT A PIRATE" and the other guy wanders around Las Vegas dressed as a pirate and gets ridiculed. He either has to come to grips with the fact that he looks like a fool (which he does) and take off the pirate costume, or he is left with the option to get angry and leave Vegas and lose his friends.
I think this is actually a GREAT scenario and would be happy if my players wanted to explore this.
You as a DM should be prepared to play a little psychiatrist now and then. If all your game is about is just go here, solve the puzzle, kill the bad-guy, get the girl/loot/whatever, that's ok... But it will be more enriching if you are actually allowing people to explore different aspects of themselves in a safe friendly non-judging setting. Your job is to present a world where people can be anything they want to be, and then cause that world to react to them in as realistic (to that world) a way as you possibly can.
Telling the guy "you can't be batman, that's stupid" (or something like that) is actually probably the less sensitive, more humiliating option here... However, if this is going to make the game un-enjoyable to the other players, you could all discuss that as a group and then maybe present "batman" with an alternate gaming scenario (suggest he find a Shadowrun campaign to join or something) where he can explore this fantasy if he is still insistent. If it is clear that no one in the group is comfortable with him being "batman" then, he just can't be batman and no matter how unhappy that makes him, you (he) can't always get what you want and he is going to figure out how to deal with that too...