I'd suggest that this would be a fine job for a home machinist that wants to help out. A new screw could be cut that is slightly oversize so it fits the threading in the wood a bit more snug and the head shaped to correctly fit the existing recess.
I'd also at that point flush out the hole with some brake cleaner and compressed air and harden the existing wood with a couple of drops of the thin water like formula CA glue to avoid future wear on the wood.
Or a more period correct solution would be to use boiled linseed oil in the hole to soak into the wood and to some extent harden and bond the wood grain into a more firm consistency. The screw should be waxed lightly about every other time it's removed and inserted as a further aid against corrosion and wear on the wood fibers. And NEVER tighten it to more than a moderate pinch.
One thing I will caution against just in case you're considering it. And that's the "just drill it out and put in some dowel" fix. From a wood working perspective this is a VERY BAD idea. I know it's common but it's still a bad idea for a screw which will be removed and replaced often. Hell, it's not even all that great for a one time fix that is never taken apart ever again. Why you ask? While it's an easy fix it puts the grain fibers of the wood along the line of the screw. So the resulting threads are ALL short grain across the thread roots and the threads WILL shear off easily the very first time anyone uses the slightest excess of torque.
For a hole that is totally badgered out and where there isn't room in the tang for an oversize screw without drilling out too much metal a far better fix would be to use a mortise chisel and take out a long "key" of wood from the stock and then form a long grain piece of wood to fit neatly into the mortise with glue setting it into place. Or with a suitable jig to hold the stock and guide the tooling a router could be used to cut the mortise. The grain of the key then runs along the same axis as the existing wood. And when the new hole for the screw is drilled the threads it cuts will be along the grain and into firmly supported end grain. Both of which will support the threads far more effectively than a long grain dowel.
OK, wood worker's rant mode off now....
I just wanted to offer this so you know there's a "better way" of making what is more likely to be a period and gunsmithing correct fix if you need to go that way.