Move? Better idea than burn the world. There is only 12k inhabitants. What's the strange idea that somehow you have a lifelong given right by the universe to live somewhere no matter what just because that's how it always was? I see that entitlement as well when people 'need' to live in a city they cannot afford; why do you think you somehow can do that if you can no longer afford it? If things change and you cannot live somewhere anymore, why does the place have to change and not you?
This reminds me of an old Simpsons episode, where Bart and Lisa both close their eyes and declare that they're going to punch and kick the air in their personal space, and it's the other person's fault if they get hit. Inevitably, they're hitting each other a moment later.
This 18th-century notion of "I can do whatever I want on my land" doesn't work, because we now understand that the impacts actually go outside of whatever boundary we've drawn. Pollution, water flow, noise, animals, insects, pollen, dust, etc don't magically stop and reset at arbitrary borders.
If what someone is doing on their patch of the Earth is harming their neighbors, of course their neighbors have a right to prevent (or mitigate, be compensated for, etc) that harm.
The area in question is not in Nauru's EEZ. It's approximately between Hawaii and Mexico.
That's a lot. Billion of people will envy them.