Assume my kaiju has just been blown open by the military, its organs and intestines spilling out.
What gas, either released from its innards or released during decomposition, could ascend and cause rain?
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The Kaiju's corpse is much warmer than the surrounding climate, and releases large quantities of water vapor (thousands of tonnes -- the Kaiju keeps the water as hot, pressurized liquid which vaporizes when the body is breached and internal pressure lost). This vapor will immediately result in a huge increase in local humidity and produce precipitation -- fog, rain, or snow, depending on the ambient temperature.
This will be very localized, of course (the clouds in an ordinary cold front are hundreds of thousands of tonnes, much too big for a Kaiju to carry around internally).
Don't expect a refreshing rain, though; along with the water vapor will go hundreds of tonnes of various very stinky Kaiju body chemicals; the rain will not only smell like raw sewage or rotting meat, it will be quite toxic as well.
Your Kaiju corpse is burning, releasing clouds of smoke full of silver iodide and other salts that act as seed particulates. This has the same effect as cloud seeding. Other alternatives are if the kaiju is full of liquified propane.
Similar fires are being used on an industrial scale in places like China to control the weather and cause rains to occur when and where the government wants.
This is a Frame Challenge
Humanity knows of precious few chemicals that can "cause rain" (or, more accurately, increase the likelihood of rain). A trivial Google search into "cloud seeding" will list them. The most popular are silver iodide, potassium iodide, and dry ice (frozen CO2).
You might be tempted to believe that since (e.g.) potassium iodide is a nutritional supplement (commonly used to iodize salt) that you can use those chemicals with impunity. Not really. In fact, in quantities that could cause rain (even in something as large as a Kaiju), they're down right poisonous. Besides, I dare you to have your dead kaiju decompose into solid CO2 (never mind how it gets into the atmosphere...). Anyway...
That's all boring
Let's look at what could cause the likelihood of rain to increase. Forget whether or not there's an actual chemical that can bring these causes to pass. Think something like this:
Marshall Pentecost, Dr. Geiszler thinks he's discovered why it rains when a Class 6 Kaiju dies.
Well, get on with it, what causes the rain.
I wish it was that easy, sir. Newton insists on calling it "Gottliebium" because, and I quote, "it's always raining on our parade." All he'll tell us is that it's a gas released during decomposition that exacerbates ___ insert mechanism here ___.
That's Newton. Was he kind enough to tell us how to make it stop?
Um... yes, sir. He's perfected a counter-agent gas. He won't tell us what's in it, either, but he has suggested a name....
Let me guess. Newtonium.
Uh, yes, sir. He said something along the lines of, "the sun always shines when Newt's on time!" and then he started playing air guitar.
If you're going to worry about the specific chemistry that's causing the rain, you'll be forced to explain how that specific chemistry is in your Kaijus in the first place, and that rabbit hole runs very, very deep. Don't do that. Pick the mechanism, ignore the chemistry. That's my frame challenge.
So, what mechanisms can increase the likelihood of rain?
Water is naturally cohesive, which is a fancy way of saying water is sticky. If enough water is in the air it naturally bonds together to form droplets. Therefore, your gas increases the cohesion of water.
Water condenses when it gets cold. So your gas could reduce the local air temperature.
The effect of #2 can also be brought about by a low pressure zone. Air rushes in, collides in the center, then some of it forces its way into the upper atmosphere (where it encounters colder temperatures). So, your gas can cause a regional low pressure zone.
If you really want to challenge people, when subjected to a strong enough magnetic field, water is repelled. Yes, yes, yes... it must be a whomping strong magnetic field, but we have Kaiju, right? Your Kaiju is magnificently magnetic, and the gas is the source of the magnetism. When released, it pushes water away from the area of the Kaiju, causing it to momentarily "bunch up" around the perimeter. Which means it rains around the Kaiju, but not on it! (And the Kaiju Cultists go wild!)
You know what our atmosphere has? A lot of free oxygen! About 21% of our atmosphere is free oxygen. And do you know what oxygen needs to become water? Hydrogen! But does that mean your Kaiju is emitting hydrogen during decomposition? NO! Think "Hindenburg" Besides, I've watched hydrogen squirted into the atmosphere (in small, controlled amounts...) and it doesn't precipitate into rain (at least small amounts don't). So what do we need? Well... the Hindenburg... but in a controlled fashion! When you burn hydrogen, what you get is water! So when a Kaiju dies, it burns! Not explosively, but fast enough to inject water into the atmosphere.
But you do have a problem
Any and all of these mechanisms will increase the likelihood of rain. None of them will simply cause rain. I don't believe it's possible to guarantee you'll always have rain. At least not scientifically, which is why I'm advocating that you care about the mechanism and forget the chemistry.
In your world, there's always rain when a Kaiju dies — and we know the mechanism that's causing it.
And if you really do want to know the chemistry, we know that, too. It's Gottliebium gas.