I would think the ascending kaiju would cause a tsunami that could be detected by all the ocean sensors the world has placed around right now.
Nope. As I noted before, big as they may seem on human scale, they are still small compared to the ocean. Kaiju are not big enough, and move at only modest speeds. If it were asleep inside a rock in the asteroid belt, and fell from space and crashed into the ocean, that would probably be noted. But just waking up and swimming around? No. You've got a better bet with sonar nets.
And I still say that you can't count on bunker busters. This thing must be made of what we'd consider "exotic materials" to merely stand and walk.
As for fighting it, I highly doubt anyone would nuke the creature as anything but an absolute last resort. And probably not even then while it was in Japan. If it were to finish with Japan and move on to China, maybe China would nuke it on Chinese soil. (I think the US and Russian would also be willing to nuke it on their own land, at least as a last resort.)
Assume that it is stomping around in Japan, and conventional weapons haven't made a dent (as is usual for kaiju), and the thing started stomping toward China... I am not at all convinced that China would not, in the name of saving their own people, try to turn it into a crater *before* it reached Chinese soil - and that'd probably mean getting it while it was sill on land in Japan.
Remember, the thing is a walking natural disaster. It levels entire cities. The city it is in has probably already been destroyed. Sure, the nuke would kill more people, and make the real estate unusable. But, they can work out the diplomatic niceties later - it isn't like anyone would mistake it for a prelude to war - the real target is pretty darned obvious, and its potential threat clearly demonstrated already.
And it would be a competition between nations to learn from the creature. Everyone would want to examine the creature's carcass, and it might even lead to massive war between those competing for access to it.
That assumes there is a usable carcass. If you need to nuke it, it (hopefully) vaporizes. If not, it ends up highly radioactive, if it wasn't already. You don't need, and probably wouldn't have, a war over it - the expertise (and raw number of people) needed to deal with such a thing doesn't rest in a single nation (even the US). It'd be scientific consortium time. And I don't foresee any blame being thrown around for its creation - see previous note about its material properties. Nobody on the planet could currently engineer the thing, and everyone already knows it.