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Source: https://writers.stackexchange.com/a/34897
License name: CC BY-SA 3.0
License URL: https://creativecommons.org/licenses/by-sa/3.0/

Exposition as dialogue is a good way to keep the story moving forward, but only if it still makes sense as dialogue. You can't have two senators discussing the name of the state's capital, but you might be able to sneak it into a discussion they're having about a recent gerrymandering ruling.

In your example "Why do you say you're from Different Big City?" seems like an odd question. If they accepted GotR's explanation of where he's from they are more likely to have asked typical follow up questions related to the place or, in your sketched scenario, give them a mysterious warning to the effect that they don't want to advertise that fact. If they didn't believe that he was from Different Big City they're essentially asking "Why are you lying?" which doesn't always work with close friends never mind suspicious casual acquaintances.

Maybe your fleshed out version makes perfect sense, but my best guess is that the dialogue doesn't have a purpose beyond providing your exposition (can you give it one? Is this something the readers would care about?) and/or doesn't make sense for the characters in that situation.

Original score: 4