The problem as I see it is that S.Korea has lots to lose and very little to gain by any conflict with N.Korea. SK has a successful, stable, functional, and wealthy population. What do they have to gain from a war with NK? Nothing. There is nothing in NK that they need or (really) want. The reunification idea is essentially a dead letter to the vast majority of the SK population today (remember: the country divided 60 years ago). Further, while the USA is the obvious target of a terror cell+nuke device, SK is not. So the medium and long term interests of the USA and SK are divergent. Thus far the SK policy of low-grade appeasement towards NK has been successful. Why should they change? The recent sinking of an SK destroyer and more recent shelling of a border island do not change the calculus of the SK overall strategy. These are pin-pricks.
Could the USA unilaterally adopt a blockade (still less embark on a military option) against NK without SK agreement? I don’t think there is any chance of such a thing. We need the SK approval and they won’t give it.
As a result, we will do nothing, except hope. BTW: I presume we have threatened NK with retaliation if a nuke goes off in the USA. Whether the threat is believed by the Kim “mafia” is an open question…
