Now, what would happen if we cool it to feed it/hasten its maturation? Can we do so without catastrophic consequences for us/the region/the planet/the system? If we get it to mature, Steve could potentially track its movement as it returns home, and it could allow for better studying its life cycle. Remember, our primary mission here is studying the eater, especially as a potential weapon against the UWM. Killing it properly is kind of a side quest, isn't it?
We know it throws out a bunch of shards if it matures - we might consider cooling it and then letting it do so, then trying to collect the shards for study unless they're alien ghosts themselves or something.
We also may want to go and examine the pit we're supposed to kill it in regardless of mission success and try to figure out why it's so suitable ('there's no shade' doesn't fly, since there's no shade in the middle of any other patch of desert), because I suspect there may be alien garbage behind that as well. In addition, there's probably at least one Eater of Cold that's been killed there in the past, otherwise there'd be no confirmation of the technique - remains may be left behind.
Perhaps the old Eaters still live there, or at least sleep there, and when somebody's thrown into it, they just make a break for the body, thirsty for coolant? Even if there's an Eater in there already? If the fragments can lay dormant in the desert for a load of years, the Eater itself would probably endure as well.
Also, if ex-Dern doesn't feel like moving in the next couple of hours, we could ask Steve to get us a camera with a stupidly high framerate, then pressure a local into shooting the host dead and filming the transfer process, so that we can maybe get a picture of it. Maybe kidnap some children who don't know any better. It moves slowly enough that we can barely register it if I read correctly, so it should work pretty nicely unless the ghost in the body wises up and tries to kill us all for some reason.
I wonder if it knows the difference between an organic lifeform and a reasonable fake - say, a robot body. Robot bodies are obviously poor in terms of temperature buffering, but what if we made one out of some sort of material that has a heat capacity comparable to that of water, then attached it to a remote-controlled wheeled platform so that we can make it move without the body's input? We could even put a gun inside it and have it shoot the eater, see if it knows what's up and if flesh is the important factor here or simply water.