Howard E has done some of the best work on BYC on dealing with rats. You really, really, really, ought to listen to his advice. Search his back posts and find one called Rats 101 or something similar, it is the definitive guide for stopping rodent infestations.
Not all feeders allow chickens to flick feed out, my feeder has the 1/2" inward lip that stops 95% of the birds from raking feed out, then the feeder lip extension stops the rest of them. Most feeders are made pretty to sell and easy for the birds to use without a thought to feed raking. My feeder is made to stop rodents and wild birds and feed waste. Like you said, feed on the ground isn't helping.
Limiting the feed isn't the answer? They don't eat much? Oh my.... Limiting the feed is the only sure way of eliminating rats in a coop. And they eat a lot. Many reviews on our feeder mention a huge reduction in feed used, two, three, even four times if it is wild birds that are the problem. If you think you have ten to twelve rats you likely have twice that number and dozens of babies in the nest.
Here is a clue for you that you wrote yourself. Your belief is that when your neighbor lost their chickens the rats came to your coop. Ya think? Seems like feed elimination at your neighbors (unless you think they kept feeding after the chickens were dead) worked like a charm, the rats were smart enough to move and find another coop that was willing to feed rats.
Another thing, 200 feet from a river... rats and mice are known to live within a few dozen yards of their food source and no further. Rodents do not have big territories because the most dangerous thing in their life is traveling searching for a source of food. The predators pick them off, birds of prey, foxes, coyotes, domestic cats and dogs, snakes, all sorts of animals will kill and eat rodents. A rodent that travels 200 feet often isn't going to survive. Rodents leave a stream of urine as they travel and some birds of prey can see in different wave lengths and that urine stream is like a neon sign.
The amazing thing is that you are thinking of spending hundreds of dollars on a concrete slab and burying wire. The rats will love that slab to tunnel under. A darn feeder will cost a tenth of that and take a tenth of the time to install and will actually work.
But one thing, please, please, please do not buy my feeder. In fact I am going to watch feeder sales from Maine in the future. You wouldn't be successful with a feeder as your mind is made up and no one is going to convince you to follow the instructions. I am going to recommend the grandpa feeder and buy it off Amazon so you can return it.