DC Comics' Death Note's Light Yagami is an equally brilliant killer with some supernatural abilities on his side, so it's far from a sure thing. Comparing these two characters and some of their canonical feats and abilities is the only way to find out.
Batman is a character with a long and colorful history, so he has a bit of an advantage in the "feats" department. However, an important part of this question is determining exactly which incarnation of Batman is up against Kira. For the sake of simplicity, it seems using the modern Batman from the comics is the best way to proceed; other eras of Batman may fare a bit differently, however. Batman's abilities include thorough physical and martial arts training, vast wealth and resources at his disposal, and a genius-level intellect that's been reinforced by the best schools. Batman's curiosity has led him to learn as much as he can about a wide array of fields, from forensics to magic to tactics. While this comparison will look only at Batman and Light, rather than their allies and extended teams, the ability to cultivate strong relationships with a wide variety of people is another critical ability of his.
There are two factors that are likely to be most important here, though: Batman's mastery of fear is a unique tool in his arsenal that could work particularly well against Kira, as fear proved to be one of the most effective ways to provoke Kira into making a mistake.
For Light's part, he's a fairly ordinary high school to college-age student, depending on exactly when in the story of Death Note he'd be from. characters like Misa or Ryuk--something that was essential to getting away as long as he did.
Batman and Kira, in a universe where both existed, would quickly fall into a similar game of cat-and-mouse as occurred with L. Batman would undoubtedly notice a string of criminals dying suddenly of heart attacks, particularly if any of his rogues gallery were targeted. Batman's secret identity would keep him safe from any immediate reprisals. Batman can deploy with his money certainly exceeds what L and the task force had access to, which may help accelerate the timetable.
In an ideal scenario for deduce Batman's identity that it's certainly possible--or even likely--that Light could as well.
Where this gets tricky is with a certain sticking point in the Batman fandom: which identity is the real one? Some say the masked Batman is obviously the alter ego, while others argue that his concept of vengeance and desire to stop crime is so intrinsic to who he is that Bruce Wayne is more of a "character" than Batman is. Even if Light got Bruce Wayne's name and face... it might not actually matter. If Bruce really, truly does see himself as Batman and think of himself that way, then wouldn't "Batman" be the name that needs to be written in the note? Of course, that wouldn't help other people associated with Batman and/or Bruce Wayne, should their identities also be revealed. Kira may well take out many of Batman's friends on his way down, but sooner or later, Batman would find him, and he has the tools to restrain someone such that writing is impossible. If Batman destroyed the Death Note and Light lost his memories of being Kira, he might be in for a very long stay at Arkham. And that leaves one last interesting question: could Light ever be rehabilitated, if he survived a final showdown?
It's hard to say, but Batman would undoubtedly try.
