Yeah, it's more natural to say, "gets its exercise." This even back-translates properly.
Also, how do you distinguish between a small dog and a small-sized dog/breed of dog which is a definite size bigger than miniature/toy but smaller than "medium(-sized)"? 작은 소형(小型/小形)/품종(品種) 강아지가 운동을 해요. For some reason going through Japanese this part gets lost in the translation . . .