While you are working on a way to trap and kill some of these guys, may I suggest an additional step? If rats are in your run, most likely they are there for the feed, plus they probably have built tunnels beneath it.
If the infestation is that bad, perhaps consider capping the run surface with tight fitting concrete pavers (or even concrete) to create an impervious barrier for them. Then find a way to tightly and securely store your feed so they can't get at it, and perhaps use the bucket feeders or some such that tries to minimize waste on the ground. In other words, starve em out. Not easy to do and there will be some who hang around and you work on those as you can.
Interesting that when you read through the poultry husbandry books from 100 years ago, about the only predators and such they mention are rats, parasites like mites and lice, and hawks. Rats were the headliners. Their solution was to use concrete or rat proof wood floors in coops and to keep yard dogs and rat terriers around to be on the constant hunt for them outside of the coops and houses. Once upon a time, almost every farm had a dog or two running around outside and they were not pets. If your dog wasn't a good ratter, you got rid of him and got one that was.
I like the idea of using pavers! Of course it's not possible for me to keep the area completely clean without nightly sweeping up, which would drive me crazy.
I bought a "rat proof" feeder a few months ago. This one:
http://thecarpentershop.net/mediumplywoodchickenfeeder.html (The money we've spent on keeping these 8 chickens is sickening.) But they still beak out food all over the ground. The food bag is stored safely and has never been discovered or disturbed by the rats. Since a couple of my hens absolutely will not eat from that feeder (and really, who cares since it does leave a mess on the ground), I also have a hanging feeder with the little slots to make it harder for them to beak out food. Maybe I should take that one away an hour before dusk and see if the hens will "clean up" their own mess. Something to try, perhaps.
And yes, Howard, they have burrows all over the run, the garden, and even a few elsewhere on our 11 acres (I leave the latter to the owls and hawks). We don't have snakes here so I'm pretty sure those are rat holes. Our garden situation is a "chicken moat," as seen on YouTube. They've burrowed in several places I know of, and my response has been: 1) pouring various things down the hole (ammonia, vinegar, bleach) and/or going at the spot with shovel and pitch fork.
Your comments about the poultry books were fascinating. Thanks for that.
MY "MORNING #2" UPDATE: not a single corpse in the water, and my "test smear" of p-butter on the rim of the bucket is untouched as well. It's funny, because I usually see rats in my flashlight beam when I take my first trip of the day out to the yard to let my dog out. There were none. None yesterday either. I found a couple of rat turds in the run, but not the usual amount. I wouldn't put it past these smart creatures to sense the grim reaper nearby.

It has turned chilly the past couple of nights (low 40sF instead of the 50sF we had been seeing). Not sure if that makes a difference.
One thing that the initial failure of my trap has done is that my heart is hardening a bit. The first night I kept waking up, picturing a sweet little Disney-like mouse struggling for hours to stay afloat. Then on Morning #1 I was so afraid to look in the bucket that I made my husband do it. Last night I slept quite peacefully and was only a little nervous about checking the trap, and actually felt some disappointment to see it empty. We've only lived in "the country" for 2 years. Perhaps in 2 more I won't care how much they suffer if I cannot even harvest a single tomato (as happened this summer - gr!).
This coming week I shall do my fall cleanup in the garden and really dig through the areas where I suspect to find burrows. Thanks again, everyone!