If vigilantism was a sign of being an antihero, then spiderman, green arrow, and almost all heroes are now labelled anti heroes.
I'll not post a long ass reply but all I'll say is that batman is most definitely not an anti hero.
In literature, an antihero[1] (or antiheroine[2] as the feminine) is a protagonist who has no heroic virtues or qualities i.e.: morally good,[3] idealistic,[4] courageous,[4]noble.
Not having those vitues does not mean you can't save someone or have a code of honour. Many villains have codes of honour. I then refer you to batman who possess all of these things, abiding by a moral code that he only came close to breaking. This was when he decided to use a god gun. The circumstances of this was when Darkseid, a god, was taking over the world.