Mike
In the good old days before the all wise British Government banned pistols (now only the criminals have pistols) I had a 357 Ruger blackhawk with 8 barrels. I actually had authorisation to use it for shooting pests with shot. I understand shotshells were available in the States but over here I could never get them. I used 357 cases rather than 38sp as you could get a little more shot in. Even then it was probably only that of a 9mm garden gun so it was a very light load. It was Ok on the rats to 10yrds. I used No9 shot as that was the smallest I could get the little .22 dust shot is around No12 I understand. I used beer mats to cut the driving wad from again to save space and a thin card to hold the shot in. I sealed it with either silicone or sealing wax. I understand that in the states you can get domed plastic caps to increase the capacity a little.
I cut down on the powder from a bullet load as that takes up a great deal of the case capacity I cant remember now how much I put in but it was probably half.
I am sure with the thin wad and reduced powder you will have very low pressure and the 357 is going to be shorter than 1 so it will be a light load for the gun.
I just looked at an old 38 case and it looks like it is .376 so that is just a little slack but you do need to be able to drop it in easily. I would guess if you use brass it will fire form to the chamber the worst that can happen is the case splits and you loose pressure. The internal measurement was around .336 so ideal match for the bore.
If the rim is a reasonable fit then I would go for experiment a little and have fun but dont hold us responsible for the advice!!
John