I bought iPhone while I’m stationed out here in South Korea, and I am having a lot of trouble adding AppleCare to the phone. I was told by the Apple rep to change my address and iCloud back to the US to be able to add AppleCare and that has not worked and I’m not sure what to do, but take the iPhone back. I’ve been working with them for two days now.
AFAIK if you bought it in SK then you can only purchase AppleCare for SK. Do you have a Korean address? Is your Apple ID account set to Korea? You might need both of those to be true. Whatever the case I’m pretty sure changing your billing method back to USA is not the solution.