Neighbor of mine had 2 "rat terriers"....and lots of horses, so I assume lots of grain and hay in the barn....I think there were really Jack Russells, but we always called them "rat terriers" cuz they LOVED to hunt and kill rats. Yes, they are yappy ankle bitters, but they did the job and didn't spook the horses...
If you stop throwing the rats where the cats can get them easily they will have to hunt for supper. Add 2 dogs (cuz they are pack animals) and keep the owl and the rats will be bothering the neighbor in no time.
Get a large bucket to throw the dead rats in and take them out to the back 40....feed other wildlife that might also become a predator to the rats...at the very least you'll put the feild rats on notice that they are on borrowed time if they step in your barn.
Why no poision? There are prebaited poison set-ups that conceal the bait so nothing but rats and mice can get at it. Old farmers say that the rats eat the poison and go out of the barn looking for water and therefore die outside. (Of course I have had them die in the wall of my house, but that's a whole nother story)
Given that rats breed at 60 days of age and have 5- 10 pups per litter if you don't use poison you might never get rid of them.
And I agree with one of the other posters, you would be surprised how much feed you are losing to the uninvited wildlife....rats don't eat where they find it, they take it back to their nest and hord it. My "rat terrier" neighbor got the dogs when she lost 25 pounds of feed in 2 weeks to uninvited wildlife. And she thought that her feed bins were rat proof.