Your primary concern will be neck tension since they are 0.003" smaller than what the brass is normally designed to use.
Accuracy could be just fine. I've shot .355" jacketed bullets in a Ruger Blackhawk which has a .3577" groove diameter and it put 24 rounds into less than 1.50" at 25 yards from a Ransom Rest.
Against the neck tension problem, I was thinking (a) it may not be as necessary to flare the case mouth and (b) I can always give it a very light crimp.
Accuracy-wise, I would be prepared to take my chances. This would be purely a practice load, with serious work being done either by factory ammo or .357-specific reloads.
Some of my bullets are 125gn factory-lubed lead. I know it's probably unwise to drive those at .357 velocities, but .38 Special or 9mm Luger-developed loads should be safe in a .357 case and gun, and I theorized they might grip the rifling a tad better.
With jacketed bullets you won't easily see the effects of 0.001" undersizing. As you get still smaller, the bullets can't easily bump up to fill the grooves, so the bullets tend to go through the bore cocked at an angle. That moves the bullet center of gravity off axis. This results in some dispersion, but with pistol bullets it may be rather small. More of an issue may be muzzle blast playing off the slightly tipped bullet base, and thus imparting some lateral drift. About all you can do is try it with your particular bullet to see what happens.
I would think it would be fine with FMJ ammo. however hollow points would probably perform poorly as the extra velocity would most likely cause them to over expand and possible come apart in a target.