I've always believed that Lawful describes a character who sticks to a personal code of ethics, is consistent in what he says and does, and expects consistency in others. It's not necessarily a willingness to obey laws and customs (and if the PHB says that, then I disagree with Paizo). Chaotic would describe a character that acts randomly and enjoys breaking systems. The evidence that supports this opinion is that different societies will have different sets of laws, some contradictory, and a character can't be expected to follow all of them.
Anyways, if the authorities here are benevolent, then the monk concept is out (unless he's LE, but I don't like playing evil characters). However, I don't think it's possible to classify an organization with an alignment - it's a property of individuals. Further, the good intentions of authorities can often blind them to evils that are committed by their minions on the street. The mayor thinks she's a champion of justice, but meanwhile the city watchmen are beating up minorities and detaining suspicious people without due process. And without evidence of benevolence, the term Lawful Good becomes merely propaganda in the eyes of the outsider.
Anyways, I'll just do a straight up CE rogue.