There are two main places where the square-cube law hurts large creatures.
One is the problem of moving things around. How do you get nutrients and chemicals to different parts of the body? The problem shows up the fastest with oxygen, since that's needed for pretty much everything. If your leg isn't able to get oxygen, it won't be particularly useful. (This also applies to any other chemical replacing oxygen; I'm not aware of any hypothetical creatures which don't need to move reactive chemicals around their bodies.)
Humans use a centralized system, where there's one incredibly-complex system in the middle of the body (the heart and lungs) which sends oxygen everywhere else. But this gets more and more difficult as the body size increases, since that single pump is pushing more material over a greater distance.
One alternative is to use a distributed system: effectively, the different parts of your kaiju's body breathe separately. This is sort of what insects do, with gas exchange happening all throughout the body rather than in a central organ. However, this requires that the gas exchange mechanism be both simple (to allow for multiple instances in the body) and efficient (to actually work). In Earth's atmosphere, insect respiration is unable to support anything more than about 3cm across. But higher oxygen levels in prehistoric times made it significantly more efficient, allowing insects to grow significantly larger.
So if you increase the levels of oxygen (or whatever equivalent chemical) in the atmosphere, that makes one of the barriers easier to overcome.
The other problem is, plain and simple, support. Strength tends to grow as the square of length, while weight grows as the cube. So scaling something up directly would make it weaker and weaker.
But weight can also be offset by buoyancy. Sea animals on Earth can grow much larger than land animals, because the buoyant force of the water around them can offset their greater weight.
So if you could greatly increase the density of the atmosphere—by multiple orders of magnitude—that would provide somewhat more support for the kaiju's body.
However, at this point, your planet would look very different from Earth.
Completely changing the properties of air would affect human life tremendously as well: for one, in air dense enough to support kaiju, you could effectively "fly" or "swim" in any direction.
So since you mention alien design, the best choice is probably to rely on a hypothetical future material being arbitrarily strong to support the skeleton, and hypothetical future power systems being arbitrarily effective to keep it all running.