No stainless. NEVER.
Call Sinclair's, or another place that sells accuracy supplies, and order some benchrest brushes. No steel core, etc., and they work very well. Cost about $1 each, so don't worry about buying a whole dozen. While you're at it, get a jug of Butch's Bore Shine, and a bottle of Sweet's...
Clean before firing the first time.
3 wet patches, brush 10x with wet brush (clean brush afterward), 3 wet patches, dry patch, patch with Sweet's, and if you see blue/green, repeat cleaning process. Fire one shot. Repeat 5x. Repeat 5x with 3 shot groups. Repeat 6x with 5 shot groups.
Use a bore guide, or plan on throwing the barrel in the trash.