I know of the popular Bruce Wayne, an African-American one and even a Russian one, but has there ever been an Asian portrayal of the Bat? (Excluding the whole Chinese Justice League which is a more recent addition).
Yes! There are at least two (and a debatable other).
Robin Annual #3, "The Narrow Path" (1994) features a Japanese Robin trained under a Bat Ninja (not to be mistaken with the 2018 animated Batman Ninja, who's from Gotham). Unfortunately, this ninja only appears (and dies) in this issue, so not much is known of him.
Granted, his armor is more reminiscent of those of the samurai, but he's described as a ninja.
Batman of Shanghai
Batman of Shanghai (2012) would also qualify.
It was made by a Chinese animation studio, Wolf Smoke, and was included in the DC Nation shorts. It also features Chinese Catwoman and Chinese Bane.
It's apparently featured on the new DC universe website, but I can't access it since I'm not in the US.
A bit more debatable, but Damian may qualify. He's born to Bruce Wayne and Talia al Ghul, daughter of Ra's al Ghul, who depending on the versions is either Egyptian, or from somewhere in Arabia.
There are a few comics which deal with Damian becoming Batman. Up to oneself to decide if that's "Asian" enough, though it's highly possible that Damian has an US citizenship and not an Asian one. ("officially" speaking)
There have been a few Japanese adaptations of Batman, but one has sort of crossed over into the "official" US continuity.
In the 60s during the "Bat-Mania" era, there was a Japanese manga starring Batman, by Jiro Kuwata.
It was wildly unique, and featured new villains like Lord Death Man, Dr. Faceless and the Human Ball.
Grant Morrison and other writers have folded some of those villains into the DC Comics adventures, with only the most hand-wavey of explanations as to how and when they connect with the rest of the stories.