That looks like a standard wood screw. Since you already know the head width and size openings/holes in the patch box, you should be able to figure out the length from either the picture or depth gauge, or how far a small wire will bottom out in the hole. From that you are likely to be able to determine threads per inch. My question is whether this is an original rifle or a reproduction. Either way, going to Fastenal.com you are likely to find an appropriate sized screw. Most likely you will need to specify "unplated or plain finish" to the screws. If a store is not near you, you can have it mailed to your house.
Not sure what you mean by right period manufacture versus "modern junk". Most likely people selling screws claiming to be "period" are simply ordering from Fastenal or another manufacturer in bulk and reselling the screws. Fastenal makes and sells "period" fasteners to the Park Service, and makes all sorts of square headed, and other old-fashioned fasteners that you don't see in the local hardware store.