357mag will work fine for fur hunting, assuming it shoots precisely enough and you can manage the trajectory at your given range well enough to connect.

It's a big bullet, moving slowly - it'll find its way through light bodied varmints without issue. You do NOT want an expanding bullet if you're putting up fur, with the exception of coyotes. At really close range, a big bullet will make a mess of a hide, but give it about 50yrds to shed speed and it gets pretty well behaved. Out of a scoped rifle, on varmints, if you can connect at 200yrds, it'll kill. I used a Win 94 Trapper (about the only time you'll hear me admit to enjoying a Win 94) for many years as a feedlot rifle, and did well with it.

There's better stuff out there, the 22 Hornet being one option, but the 357mag in a rifle barrel isn't a bad choice.