If it is only one rat you might catch it in a trap but usually you see one you have a dozen. They wise up quickly to both traps and poison and the effectiveness falls off quickly. And yes, the rats natural predators might well eat a poisoned rat or several and you kill off the natural predators and make the entire eco system out of balance.
First thing to fix is what is drawing the rodents to your coop, you are feeding them along with the chickens. Get a good treadle feeder, get your bulk feed in metal barrels or metal trash cans with lids, and clean up an debris that give the rodents cover to and from the coop so the natural predators have a shot at catching them.
If you do this you will rarely have to resort to poison or traps however the traps become a lot more effective once the rats are starving.
Treadle feeders come in all sizes and price ranges. The best have inward swinging doors, spring loaded doors, narrow and distant treadles instead of a wide step, and pay close attention to the number of negative reviews. Those are the folks that actually had a rat problem, the others just wanted a chicken feeder.