having looked at the photo, I would dig down around the edge about 10 inches, run an apron of hardware cloth around the edge, that goes down and then flattens or have it go at a 45 degree angle away from the coop/run all the way around. to make the attaching easier, you can use metal trip and drill holes in it every 4" or so and attached the hardware cloth pinning it between the coop and the trim. I would then back fill with dirt and possible top it off with 5/8th's gravel, it should deter rats and everything else. you are fortunate that you have so much concrete right near the edge, I think that will help assure you never have an issue.
the other approach would be to pour a 3" concrete border, filling in the space. that would take some mixing but might be the cleanest look with the least digging. In that case I would run ground contact rated pressure treated 2x4's (you can tell ground contacted rated by perforations in the wood that allow the treatment to soak in deeper), around the foot print, whick will raise the coop off the ground and protect the thin wood at the base. I would pour the slab to be flush with the 2x4.
