First, look under the butt plate/pad, to see if the observed finish is the real wood, or a faux finish.
If it's a faux finish, I would advise leaving the stock alone, other than cleaning/waxing.
If the finish isn't faux, FWIW, the Browning standard finish is an epoxy, and can be a
BEAR to remove - I would suggest an epoxy-effective chemical stripper like CitriStrip instead of sanding, then raising the dent with a hot iron held on the depressed area with a wet cloth between (for steam).
That's only if wood fibers are not broken, which would indicate the need for a filler made from a mixture of wood glus & gunstock sawdust taken from under the buttplate/pad.
I would only sand the repaired area.
For staining, I use Johnson's MinWax Black Walnut, dried at least overnite before proceeding further.
To replicate the factory finish, a spray-on epoxy finish can be used, if you're confident in your spray painting ability to leave no runs.....................(mist, dry - mist, dry - etc, etc)
.