Okay, so the base theme of BvS was a dark one. Few points that I would like to put some light on -
- Superman first appeared from the sky and created mass havoc while fighting Zod and his army.
- Superman made so much destruction that people started taking him as a threat more than a hero or anything.
- There was havoc everywhere, police and government not having the peace they want.
- Batman doesn't believe in Superman, right from the time he saw his casually demolishing buildings. It is shown multiple times that Bruce believes that Clark shouldn't be trusted; that me might as well be our enemy.
- Bruce doesn't have faith in the system and police anymore. He make his own justice for the wrong things that he sees.
Considering the tension that is constantly built up in the movie, the state of mind people are in after the alien war they witnessed recently, the team avoided to directly mention the name Batman.
Batman and Superman are the names of the hero, and the whole plot was built up on tension, revenge and anger; where the necessity of mentioning anything heroic wasn't there.
To add weigh on this, the writers even added other names for Batman and Superman as well, like - Caped Crusader, Bat of Gotham, Man vs Alien, Red Cape and others.