Most of the times we see Batman or Bruce Wayne in L&C Fanfic it is the one we are comfortable with from the comics.

In the new DC movieverse, which I dislike, the Batman brands people and kills people.

I think the Superman of L&C would have a very real problem with him.

How about a L&C tale where Superman has to deal with this anti social judge and jury Batman?

The Superman in the DC movieverse is willing to level large parts of a city with who knows how many deaths. So they are both evil. This is why I hven't seen any of the new movies.

It's certainly a possibility. I did something similar with the current TV version of Green Arrow in my Supergirl series; he killed dozens of people when he started as a vigilante, I had Batman track him down, with the help of information from another dimension's Dick Greyson, and put him behind bars.

To be honest, I'm usually not that bothered by superheroes using extreme measures in their fight against evil in and of itself. (Murdering people is often too much for me, though, unless it's somehow justified, either as an action or as a plot point - e.g. having some thematic significance, or the POINT is a chink in the superhero's moral armor.) However, I mostly found that movie's Batman to be terribly immoral because of his motivation for wanting to destroy Superman - just because he's too powerful. Being so overwhelmingly powerful is a physical aspect of Superman's nature. To want to kill someone because of a physical trait they were born with is horrendous and xenophobic. It's the exact opposite of being a hero. Ironically, Batman was still the best part of the movie even though his morals were atrocious simply because his personality was well done and Ben Affleck did a good job acting.

Anyway, this is an interesting idea for a fanfic challenge, though if I were do it, I'd have a different angle, as you can probably tell from my different perspective.

That's pretty much the same motivation they gave to Lex Luthor (the evil business tycoon version) in the comics

If you really want to see an evil Batman, read Frank Miller's All Star Batman and Robin. That take on Batman makes the one in Batman v. Superman look positively gentle.