What if it did in the 70's-80's, how much opposition would there be in the South (presumably among the military and conservative politicians)
Well, it might depend on what period in the 70s/80s you're talking about. Park Chung Hee had been re-elected as a technical civilian president in 1971, but began ruling as a military dictator in 1972, when he declared martial law and enacted an authoritarian constitution. And Chun Doo Hwan, who came to power shortly after Park was assassinated, as far as I know was never anything but a military dictator. Roh Tae Woo was also a military man, but was voted president in 1988, in an election that was probably about as democratic as it could be, taking place during the last days of a military dictatorship. I think Roh governed more-or-less democratically, though enforcement of the National Security Laws were almost certainly harsher than it is now(it does remain in effect, and is still occassionally used, albeit quite sparingly, to harass left-wingers).
So, if the question is what would the military and the conservatives think about ending the war, well, guys like Park Chung Hee and Chun Doo Hwan(and maybe Roh, depending how you regard him) pretty much WERE the conservatives and the military. Personally, I can't see any of those three wanting to end the war. If you're hypothesizing a situation where they go rogue and do a "Nixon To China" thing, my guess would be that there would indeed be elements within the military who would oppose that.
Under the more liberal Kim Dae Jung in the late 90s, the South did initiate a limited outreach to the North, known as the Sunshine Policy. Conservatives grumbled, but I don't think there was ever any serious talk about the military stepping in to toss him out. Not sure how it would have gone if he'd actually proposed a peace agreement, and I don't even know if that's what he was aiming for in the immediacy. Despite the anti-American rhetoric of some of his followers, KDJ never proposed kicking out the Yanks, and neither did his even more anti-American successor, Roh Moo Hyun.