Just my two cents here, stick with something clean and proven.
For fur trapping, an air rifle let`s say, a famed vintage Sheridan Blue Streak .20 cal pellet rifle. ( My understanding that the new ones don`t even compare to what a vintage is.) These will from experience , explode a noisy red cardinal at about , up to and including five yards. Fitted with premium open peep sights and an offset red dot laser, you can`t go wrong on five pumps and ten feet from a set, clean ear shot to prevent pellet holes. The animal will just lie down. Clean and humane. It`s more in the neighborhood of shooting tranquilizers, no loaded weapon to carry around, carry a shoulder holster for a hide out if you need protection with a magnum snub nose, otherwise an accurate extremely powerful pellet rifle for fur trapping is an excellent choice to carry, the old ones even better, some came in stainless.
A Sheridan Blue Streak will punch a hole through a cottontail at 10 yds. There are also Co2 pistols that are high powered which are maybe strong for five shots, however getting up close and personal on a trapped animal becomes a Mexican standoff at times , face to face it`s very difficult to find an ear shot. Ten feet and peep sights or a red dot. and a little patience especially on larger mammals will take allot of stress out of the work for both.
Cats and Fox are the toughest of all, it needs to be cleanly done, inside the ear or right in front of it, then sit down and let it be for fifteen minutes and don`t go up and disturb it, just let it go out on its own....sometimes it takes a little while, wild animals are tough to the last breath.