Would it be enough to take out North Korea's nuclear capability and cripple troop encampments on the DMZ enough that they can't immediately launch a massive counterattack on South Korea?
You could move the government, but it wouldn't matter. Seoul would still completely dominate South Korea. Their entire economy and around half their population is based in the Seoul metro area. It's only a mild exaggeration to say that Seoul is South Korea.
I wonder how much they'd ask.
There are better ways to deal with this. Diplomacy ... perhaps
It's not worth it to the U.S. or anyone else to invade, lose 30K-100K soldiers and then have to rebuild the country, and it never has been. The nuclear arms race serves partially as a deterrent against possible invasion, and partially an attempt to get people to the negotiating table for some resources by scaring Japan and N. Korea enough that they want to deal with this.
But if they had the capability, they still don't have the desire to use it as a first strike since that would give every country around + the U.S. a reason to bomb them to the ground.
What definition of "worked" are you using? Serious question.
"Diplomacy hasn't worked" is junk to say but "attacked no-one" isn't right either.
NK falling is considered terrible by all parties. South Korea can't absorb it. China can't absorb it. It's a hot potato no one wants.
That said, all-out massive attack isn't viable. there would be no way to disguise the political and military preparations for such an attack, and China wouldn't allow it anyway. Bear in mind the the South Korean capital Seoul is within medium artillery range of the border, and there are a lot of North Korean artillery pieces aimed at it. Even without a nuclear attack capability, North Korea could do them an awful lot of damage in ways that would be difficult to prevent.
"Oooo a present! What is it?"
"It's a T-shirt with a giant logo reading "FUCK YOU CHINA", with a red bullseye painted on the back. Like it?"
That said, if Japan decided it _wanted_ nuclear weapons, it can almost certainly produce them itself very quickly (order of 6-12 months according to most estimates I've seen). They have the raw fissile material due to their longstanding nuclear power plant program, and the general knowhow and infrastructure. https://en.wikipedia.org/wiki/Japanese_nuclear_weapon_progra... has some more data.
The problem is, they have hundreds (thousands?) of artillery pieces buried in mountain bedrock and pointed at Seoul. Even if we dropped paratroopers on Great Leader's palace and captured him in the first two hours, it is simply not possible to find, let alone knock out, all of the artillery within a few days, which is how long it would take for them to level Seoul.
...If any of them still work. An awful lot of NK's military equipment hasn't been maintained for decades and has rusted solid where it sits. So the artillery could be one big bluff, for all we know. But can we risk it, with South Korea's largest city at stake?
Update: apparently I wasn't the only person to think of this: http://www.nknews.org/2014/07/how-israels-iron-dome-could-he...