Yeah i find there are to ways to hunt the big cats, cover scent lotion and try to stalk them or stay on the horse because it shows you on the radar where the cat is in time to get a shot off. Mostly used the latter when I had to kill 5 cougars with a bow.
As to preferred long guns, I tend to use bolt action and Lancaster repeater swapping out for the shotgun when needed, though I find that the shotgun is more for people that don’t use deadeye since I am going for headshots mostly.