It's important to note that this is a false analogy. Contrary to popular belief, the Immortal Flames are not a law enforcement branch. They're Ul'dah's military, and the Brass Blades are law enforcement. In your example, it would be as if an off duty US Army officer showed up and tried to act in the place of the police, rather than calling the police to deal with it. Yes, harming the officer would cause a good bit of trouble in its own right, but you're under no obligation to regard an Army officer as you would a police officer. And he'd have no right to arrest you. The Army fights our enemies, whereas the police protect and serve the people.
Just the same, the Immortal Flames don't patrol the city and make arrests. They fight the enemies of Ul'dah without. The Brass Blades take care of Ul'dah itself, policing its streets and Thanalan's roads (of course, they're often corrupt, but so are the police, on occasion). The Sultansworn are way off in Secret Service land, with authority that trumps both, but only when specifically serving the Sultanate.
Of course, the root of the problem here doesn't have too much to do with all this, but it's worth clarifying. The problem might not have existed at all if the Flames roleplayer was aware of this. Then again, maybe they were, and their character is just shady and corrupt, trying to take advantage of their position. There are certainly examples of military sorts doing that in real life.
Either way, there's plenty more at work here that others are addressing nicely.