Yes, South Korea should have its own nuclear weapons, because it should dismantle North Korea once and for all. People in North Korea do not even know that they are suffering. A South Korean weapons build up would make it clear to North Korea that the world will not tolerate North Korea's human rights abuses.
South Korea shouldn't station their bombs like North Korea, where they made their own and shows off internationally that they would bomb the South if they didn't obey. The South shouldn't have any bombs, but they should have the resources to quickly create one when a war is immenent, just like lego pieces. And they should have a lot (Around 10 to 15 should be plentiful) so that they could scare off the North. This way, South would have a better chance to become a sucession of the unified country later on.
Nuking can't be used for country that close .And if needed U.S have nukes scattered across submarines ready for firing. But north korea plan is to fire nuke to U.S. South korea need to prepare for invasion and ready bunker but else it's ready. They should focus more on conventional controllable warfare. They're proxy urban war. Not suitable for nuclear war.
South Korea gaining control of nuclear weapons will not help anyone. If they were to bomb the North then they would be killing thousands of innocent people. That's North Korea's ideology, not South Korea's. If they do get nuclear weaponry then there will be another Cold War situation where weapons keep getting produced and no one fires, only this time it may not be such a "Cold" war.
South Korea has more power and technology and needs to be the better side. They have the right to defend themselves but weapons of mass destruction is in no way defense. You can't fight bombs with bombs or violence with violence.