They could simply let in 100,000 immigrants to fix the growth rate. I know it's not as simple as that in a society like Korea's, but 100k is perfectly manageable for such a large country. Take primarily young people with some useful level of education. They'll work, learn the language, marry, and start new families that become primarily Korean.
According to Google, Russia's rate is -0.4%, this might be the last war they can afford.
China didn't end their one-child policy out of a great love for children.
Both of these countries now have ethnical enclaves, thanks to multiculturalism. So I'd say yes.
Current fertility rate in Korea is 0.78/woman
Gen1 = 100 Gen2 = 78 Gen3 = 60 Gen3 = 46
Gen 1 - 100, Gen 2 - 39, Gen 3 - 16, Gen 4 - 6
But if they did, and for whatever reason none of those who has emigrated from Korea wanted to go back to the old country, there's a whole extra Korea just on the northern border, and they used to be the same country…
(Of course South Korea's case is very particular because they actually have an enormous reserve of eager potential immigrants who already speak the language, living just across their land border, an hour away from Seoul. The problem is that the other side of the border won't let anyone go.)
to reduce the NK situation to "they won't let anyone go" is also disingenuous. even if you ignore that the US refuses to allow any more meaningful peace so it can justify a continued military presence halfway across the globe, the US and South Korea are clearly more powerful and wealthy. clearly they are in a better position than NK to make concessions and pave the way for integration.