Playing a race in a culture that they are not native to...
Although more seriously, in-game, Ishgard is primarily Elezen with Hyur. The Drachen-clad dragoons you see in a sidequest are all Elezen. So, essentially, if you want to go with authenticity, accuracy, and appearing like a natural product of that nation, you'd want to play either Elezen or Hyur. Anything else would be regarded as an exception, and just requires an explanation in the backstory.
Something that would be pushing my immersion (and you really shouldn't care about my immersion in particular) is it if it was Au Ra or something.
Although more seriously, in-game, Ishgard is primarily Elezen with Hyur. The Drachen-clad dragoons you see in a sidequest are all Elezen. So, essentially, if you want to go with authenticity, accuracy, and appearing like a natural product of that nation, you'd want to play either Elezen or Hyur. Anything else would be regarded as an exception, and just requires an explanation in the backstory.
Something that would be pushing my immersion (and you really shouldn't care about my immersion in particular) is it if it was Au Ra or something.