Okay, but the idea of Zuko stopped from finding Azula by bureaucracy is mind-boggling. This is a person who as a powerless teenager wandered the world for 3 years to find the Avatar to restore his honor, then spent another season chasing him from the South Pole to the North Pole. But when he's Fire Lord, with so much more power, bureaucrats can stop him. A testimony either to the power of the UR bureaucracy, or to Zuko's haplessness in the face of bureaucracy (or his lack of motivation) or both.
Not really? In the case of the modulus, it makes sense because it entirely becomes a sovereignty issue. Remember, in the modulus-verse, the United Republic is founded with treaties that actually have teeth. They have national sovereignty rights that are stronger than the ones implied in LoK. For example, the other nations do not have a say in how their capital city (Republic City) is actually governed.
In the real world, nations do not have the right to just ask each other for information on random citizens for no reason. Trying to get the personal information of another country's citizens is usually called "espionage". Which Zuko would be engaging in since he really has no idea that Azula is even there in modulus. He thinks she never made it out of the Fire Nation.
In the KS!verse, this is very much a slight-of-hand legal loophole misdirection intersecting with a slightly archaic system that had been hastily modernized. (The person filling out the forms, for very good reason, filled them out as if Azula was a random, parentless war orphan. If someone was looking for it, they could figure out that the young teenage firebender taken in as a ward was actually the missing princess. Otherwise, the officials filing the paperwork are going to miss it because it looks like Just Another War Orphan.) It very much causes a legal headache once Zuko finds out it happened. He's not happy for a lot of reasons.