I don't have a problem with shooting rats with a non FAC Air rifle, Do need a bit of patience but they tend to move along the same run and often within the same time window. Their eyesight isn't fantastic ,They rely more on hearing and scent which is understandable for a burrowing animal. So staying quiet with clothing that doesn't rustle and downwind with the air rifle pointing roughly on the run the rat is often stationary or just ambling along. At night a maglight taped to the barrel was useful as well if you knew roughly where ratty was by hearing him and suddenly switching on,the night vision sight I have now makes it even stealthier. Using a compressed air rifle with a 10 pellet magazine you can often get a 2nd rat before they reach the safety of a burrow. Though I haven't seen any for a while now,usually get them when the adjacent cornfield is Harvested.
G.Harman