Mechanical or other kinds of traps, vs, poison bait, I'll take the trap every time when possible, seems they drag the bait off, then die in bad places where the dead corpse reeks something awful. My preference especially around barns, where they die in walls. More important, and its probably not so easy with poultry, you have to break the cycle with these rodents, interrupt the food source, or eliminate it, and or habitat for them to live in. We had them at the horse farm some years back, they can be a pain to deal with, can be wise to what you are doing, but I prefer the trap unless they get wise to it, then you have no choice to try poison, still prefer snap, sticky or multiple catch live traps when you have numerous mice. With the bait, some of them would be stumbling around in plain sight, easy to kill then, but not so good with customers around. What was most effective was cleaning immediately under the grain bucket after each feeding, no spilled grain, no food for them, that was the attraction.
I worked in commercial construction in NYC for many years, I've seen them take over a new building and get out of control in no time, more so after a nearby demolition and poor housekeeping on the new site, the most effective thing you can do in my opinion is get rid of the food source immediately, in this case it meant strict housekeeping, implementing traps and bait as well. In my case, I had a demonstrated reputation for keeping jobsites clean, my house until its turned over to the owner, right from the start I had an exterminator hired and on the job until the job was done, riding a new elevator while taking an owner for progress tour is not impressive, when rats ride with you, saw that more than once. Infestation does not take long, gets worse even faster.
I don't know how effective any predators would be in your case, some Cats are really good at controlling mice, some are not, too many variables for me, I want to find their trails and use choke points for traps with things they like, cashews, granola bars have strong odors, they work good in live traps for mice, which can catch multiples, without resetting, then you can disposed of those afterwards, I find these most effective with mice, though I have also put out excessive snap traps just the same, no overkill when dealing with these rodents LOL !