You are getting great advice from others. I think country gents dowel cutter is on the right track. But since your not cutting through the stock, snapping the dowel out will be a challenge.
I would use a Variable Speed Oscillating Multi-Tool, with narrow width blades. This would allow you to cut a wedge shape piece out from behind the butt plate. The grain will be in the same direction making it a great match to the missing area. As TNsailorman Save the saw dust you make.
Cut the damaged stock area smooth using a fine blade saw or table saw. Now you have 2 smooth edges to work with.
Given the wedge shape you have to work with, cut a larger piece out and drill a hole through from the thick end and into the stock itself. This allows you to peg the piece onto the stock. Pegging is a tried and true method of wood repair and building. It holds solid and will not release. Its the only way I build. Very rarely use any nails.
I use Devcon 2 part liquid steel as a bonding agent. And very little and in a smooth pattern. Clamping will be fairly easy to hold the two pieces in place until glues are set.
Then shape the wedge using “very” sharp wood chisels to mold a shape the profile. Sand using fine grade paper to desired smoothness.
Mix some saw dust with Old English Furniture Oil. The oil will absorb the wood color from the the saw dust and with multiple rubbings you’ll be able to match the stock as close as possible.
Time and patience are the key to this type of work.