Dont worry too much about killing small game during your career. Anytime you can load up the southeast map and go to buck hill area in open hunt. Equip the shotgun and 45 birdshot rounds.
Just to the left of the where you start there is a rabbit hole. Just keep your gun focused on it and shoot them as they appear. Sometimes they will come out of different holes in the same area too so you may need to look around if none are coming out of that hole. Once you run out of rounds just keep reloading the hunt till you get 200. Doesn't take long at all