Actually, Battle Royale is more like a dark comedy. The acting during violent and action scenes is way too wacky. People get shot and killed but overreact. One could argue that the movie is a satire because of its social commentary. Maybe the death scenes are wacky so that it is easily more palatable to address some of the more serious undertones about the relationship between adults and adolescents.
I saw Godzilla Minus One recently and wow, that hit close to home. My mother grew up in post-war Japan and had the whole survivor guilt thing. Nearly tossed herself off a bridge.
I like sci-fi and fantasy films, too, but most of the Japanese ones that come to mind are very well known already, so I'll include some other stuff as well:
I like Red Lion. It isn't either genre, but it is a feudal Japanese film that is filled with a surprising amount of comedy. Could be of interest to you if you can tolerate older movies. Shin Gojira and Gojira Minus One are also both good, but unfortunately don't fit in the genres you suggested either.