I wonder if your generators are shutting down before the gas runs out due to the lack of oxygen. You might try safely piping the exhaust into the house.
CO is deadly stuff. Once when I was young and stupid and willing to try about anything I had set up an aluminum foundry in my garage. Walked out one night to check on a batch that was melting, got near the furnace and dropped like someone had taken a baseball bat to my head. No warning, no bad smell, no struggling for breath, down I went unable to move . What saved me was that I fell right next to the garage door, big swinging door that had an inch or so gap under the door and some fresh air was coming in. I got a few lung fulls of fresh air and was able to move again, jammed my nose under that crack and sucked up half the air in the area, got up and got the heck out of that garage.
The mice are feeding on something. Mice generally are going to live within twenty to thirty feet of their main food source. Not likely to travel between the neighbor's chicken coop unless they have a safe pathway to avoid predators. You might talk to the neighbor and get them to track their feed use, a quarter pound per bird per day, ignore the roosters, any more and you are most likely feeding rodents or wild birds. If that is the case, talk them into getting a treadle feeder and starve the rodents out, increasing the competition for food for the surrounding colonies, they will fight it out for territory and you will have fewer to deal with.
The other thing about trying to gas or poison rodents is the smell after they die if you cannot get to them. I would check to make sure there wasn't any dog food or horse feed they can get to. Clean up the area of anything that offers them cover to and from the house so the predators can see them and catch a few.
I'm not advocating using poison but I did eventually when I had a rat infestation in one of my coops. I know that's not the same as having mice. I'm not sure why you have mice but I don't believe it has to be because of chickens. As you said you don't have any birds but you neighbor does. Somewhere they found a nice cozy place to live. If you have other pets and if the feed is left out, it can attract the varmints. I tried most everything when I first discovered rats to no avail. When I renovated my coop taking out the ceiling and walls, dozens of rats of all sizes poured poured out. I discovered several nests in the process. I bought some rat bait stations. The rats didn't care for the bait that came with the bait stations so I bought one that the feed store up the road used and the rats seemed to love that bait. The bait stations I use have a little window above the bait so I can check it often. I never found any dead rats laying around but they did have tunnels around the coops. I assume they went into their tunnels and died. As far as secondary poisoning, most pets that have consumed a mouse or rat may get sick but it's not usually fatal. Eating the poison directly may be fatal. There are responsible ways to keep pets and other critters away from the poison. I put the bait stations in a pet carrier so the only things that can get to the bait is the mice or rats. There is a patrician down the middle of the bait station that the rats have to go around to get to the bait. I keep the bait stations loaded with bait and check them often. I haven't see any rats in a very long time and the baits haven't been touched. Whatever you decide to do, good luck...
