The chickens might very well find a poisoned rodent and eat it. In addition, owls will be poisoned themselves if they eat a poisoned mouse or rat. Although many people dislike owls, they are nature's "rodenticide" and serve a useful purpose.
IMHO, sticky traps are messy and inhumane. Are you willing to check more than once a day to see if sticky traps have caught a rodent inside your chicken feeder? Who would want a dead/rotting mouse or rat polluting the feed while stuck to a sticky trap?
You might look at snap traps secured inside small boxes in the vicinity of the feeder. Chickens can't get inside or get hurt; quick and effective for mice/rats; no residual poisons around for chickens, dogs, cats, kids to encounter.
The bucket/tub feeders sounded like a great idea until I remembered the mice and squirrels I have in our barn. So when we can find the time, we'll be building a treadle feeder instead, with hopes that chickens will use it successfully, and that squirrels/rodents won't.
