I have returned! However, I fear what I have gathered will be disappointing.
I met my (future) boss while he was on a trip abroad and asked him what he thinks the Han in Hanguk stands for. Surprisingly, he had no answer ready. He eventually guessed that it probably is a marker for ethnicity, but wasn't sure. When I mentioned that I had heard that it was derived from Samhan, the other Korean in attendence seemed to recognise that connection, but wouldn't make a strong claim either. Most fascinating to me was that he did not seem immediately offended when I mentioned that it was suggested to me that it was the same Han as in Han-Chinese.
So in total the conversation just made me more confused and gave me no real answers. For now I will personally believe the Samhan story and try to figure out more once I actually am in Korea, though that's still a little ways off.
PS: Hadn't thought about a lack of nutritional diversity in Japan, but that sounds pretty good to me. Certainly should be a larger factor than living on what is technically an island.