Yep, It'll Work
Sure!
The RN's not so good an all around bullet shape as a SWC, but hey, if you have a bunch of 'em, and the nominal bullet size is within a thousandth or two of your .357, go for it. The RN does feed better in carbines, too.
I've loaded cast 9mm bullets in .38 for my Ruger Blackhawk Convertible, and at "tin plate" range, no problem. They group on paper just fine, too. I don't make a "regular practice" out of it, but sometimes I have 9mm bullets that don't fit my needs, like your situation, and I'll load "range practice ammo" in .38 Special with 'em.
Some bores are a bit more forgiving of bullets being a bit on the small size, but you'd be surprised how well this might work. While it's harder to get great results with, say, .308 bullets in a .311 rifle (AK, SKS) bore, you're close enough here you'll likely be just fine.