Rats colonize an area for one of four reasons; food, water, nesting materials, or a nesting space. The first two are the most common draws for a coop, the first one being the most likely.
So most people think like you do, how to keep them out of the coop, trying to treat the symptom instead of dealing with the actual problem. The feed is in a poor feeder. Fix the feeder and you have solved the problem forevermore.
Or you could attempt to build a rodent proof coop. Fix that door, keep all holes plugged, expect the rat to try to get in and chew holes through anything not ceramic, glass, or steel. No free range or course.
You should do more than buy a good feeder. Practice what our resident expert, Howard E., called the sanitation method. Bulk feed in a metal drum with a tight lid, clean up the pathways the rodents use to travel from their den to the coop, and then buy a ratproof feeder. In your case, the rat seems to have dug its burrow right under the coop. Or does it have an exit hole from the coop? Rats generally will travel no more than 100 feet between food and den, mice, half that. Because travel is dangerous for a prey species like rodents.
Now, what is a ratproof feeder? First where do you find one. You won't find one on
Amazon, only the Chinese made feeders like the Grandpa feeder, they can give away 35% of the sale price for commission and shipping subsidies. Really they just mark them up to $179.00 to cover the costs, double what a treadle feeder should cost. There are some Chinese made knock offs that aren't any better or worse than the Grandpa feeder, like the Rent a coop feeder, selling for around the same price, or some direct sold Chinese made feeders selling for $80 t0 $90.00, about the same price as an American made feeder that is far superior but again these Chinese sellers are only getting 60% of the sales price.
Now, what is a ratproof feeder.... it has to have several features for it to work. First is an inward swinging door for safety and for quick training that doesn't require the lid to be propped open for weeks during training like the guillotine style feeders (Grandpa and its clones). Blocking a door open does two things, first it teaches the chicken that the door or lid isn't supposed to move, when it does move they rightfully freak out. Then it teaches the rodents where the feed is located.
Second thing needed is heavy springs on the door to keep it tightly closed. Treadle feeders have a lot of leverage due to the linkage and the distance the door has to travel versus the distance the treadle CAN travel. Tiny springs are just window dressing. A door needs around ten pounds of force on the door crank by the time that force shows up at the treadle it is only three to four pounds of weight needed at the treadle. Then the door has to travel far more than the door crank CAN travel so you trade leverage there too, resulting in one to two pounds of force needed to just push the door open. Luckily that will prevent most rodents including squirrels, from just pushing the door open. So without those heavy springs on the door a feeder can never be ratproof. To use a lighter spring means severely reducing the size of the door, forcing the hen to lean way down to eat and causing damage to combs on roosters.
The third thing a treadle feeder HAS to have to be ratproof is it needs a narrow and distant treadle, not a huge honking treadle plate that can be swarmed by rats or pigeons. You want the step back far enough that IF the treadle was swarmed, they would try to rush forward to get to the feed and the door would slam shut. Avoid any treadle plates with holes punched in the step, they cut toes, no way to deburr the sharp punched holes, just an unsafe way to do things. But those wide aluminum steps are slick with poop so they do that because the step isn't able to be gripped in a chicken's claw like a narrow treadle step does.
You are right to be concerned about sanitation. Rats bring in lice and disease and other critters like wild birds bring in more disease and mites.
You will be bombarded with advice on trapping, poisons, smelly things, baking soda, hot peppers, all sorts of useless advice. But the two things that actually work are the tight coop and the sanitation method.
