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Assume the Kaiju is a six legged crustacean, 200 feet all, weighs 4000 U.S. tonnes and is being held in a massive cryosleep chamber.
With no knowledge of its behaviour, would scientists or would a government allow scientists to release it into a controlled environment for the sake of study?
Unless there were adequate safety measures in place (which might be hard to assess), That would be a clear violation of the most basic biohazard safety procedures. Whatever a 'cyrosleep chamber' is and however it got in there, hard to see why any responsible adult would carry out such an act.
Scientists are very aware of the risks and unintended consequences of their work, which is arguably why the whole Kaiju genre was invented in the first place.
If it is sentient, to communicate it will have to be awake.
Perhaps there are signs that this thing is more than an overgrown crawdad. For example, it was found in a spaceship clearly designed by and for entities of its kind. The cryosleep pod is integral to the spaceship.
You might wake something like this up in order to communicate with it, and learn more about where it came from and why. A species with the restraint and foresight to build a spacecraft and a cryosleep unit might not just go into a killing spree on awakening. It would be controversial to be sure but there would be good arguments in favor.
Wild kaiju crabs are normal.
If it were realized that there are at this moment a fair number of these things strolling along the ocean bottom, and the one sleeping in cryo is one of many, one could make the case that it should be returned to its habitat. If these giants stick to the ocean bottom, releasing the one you have back into its native habitat will do no harm, and save you the expense of containing it. If this particular one was causing trouble because it wandered into shallow water, it would be analogous to relocating troublesome bears to deep wilderness.
Also people might think it is wrong to keep a wild thing captive.
I like the idea of T-shirts that say FREE MEGACRAB.
Depends on what they would gain
The risk is high, of course, but if discovering the creature led to the conclusion that certain mysterious disasters were caused by a similar kaiju, people may argue that they can not throw away their only chance to learn more.
Containment facilities would be built, as best as possible. How much effort they would require before the awakening would depend on the need to know and the timeframe they need the knowledge in.
Desperation, greed, and plausible deniability:
This is more about story than about reason. It's a recapitulation of the classic Pandora's box scenario. Before Pandora had the chance to open the box, Epimetheus (who guarded the box) had a choice of accepting Pandora as the gift from the gods (who MADE Pandora for this reason, wanting to punish mankind for Epimetheus's brother Prometheus and the fire thing). Science is thus Pandora, and releasing the evils of the world is unleashing the monster.
If it was a government or scientists with ulterior or desperate motives, then sure. Think North Korea threatened with nuclear war. This thing might kill them all, but if it does, the world burns with them.
Or maybe they plan to kill it and study the body and stasis chamber. They're for-sure totally certain that the bunker-buster bomb they have will finish it off, then they can get all that alien tech.
Maybe they don't care if the thing gets loose. If they still have the tech afterwards, then the monster wandering around is CLEARLY not their fault, it got loose because (insert enemy here) went and deactivated the containment/stasis field in an act of sabotage.
You don't even need to tell us your plotline. We can all see it coming. Godzilla, Jurassic Park, Pandora. The scientist know these stories, too. Make them ignore the consequences for the gains, and they'll act. Or make them see the consequences and they'll try to stop it. Your choice.
That creature would not be able to sustain its own weight even in water. It would be very safe to try and wake it up.
The creature could be used for biological research in much the same way humanity already uses whales. There is an age old conundrum that science is trying to crack: what is the most delicious way to eat them?
Mandatory SMBC:
(Source)
If you want a non-boring answer, you should use magic rather than biology.
