I found the the answer to my own question. I asked "Was Bruce born ambidextrous?" According to one episode of the Batman (TV series), the villain Egghead used the fact that Batman was RIGHT-HANDED to deduce his secret identity. But even so, I'm sticking to my convictions that Batman is ambidextrous because if that happened once, I wouldn't let it happen a second time.
Further evidence: carefully watch his hands: