Oregon45,
Very nice old drilling. I have a great respect for these guns, I have a very similar gun, but of 1939 proof marks. The action of the gun is much more the style of late 1800 to early 1900. I am thinking this gun may have had a rebarrel and reproof at around that time. My action and configuration is very similar to yours.
I have researched these 9.3x72 / 16ga drillings and spoken to collectors and other owners. The 9.3 barrel diameters can vary from .358 to .375. If yours is the normalized case, the bore can still be of various diameter. Chambers can vary alot, but the typical ammo available from S&B will chamber and still function pretty well. You can notice the bullet configuration has a fairly short bearing surface to the bore. The bullet diameter is .366. It will work well enough unless the bore is oversize. Headspace is also variable in the older guns. It may not be very apparent unless you have ignition problems. You can pean one edge of the rim to make it thicker and put the head against the breech. This prevents the case from moving forward when the pin strikes. Mine is pretty much in the Normalized dimensions. The barrel is .366 and the chamber tight. I may have a little bit of headspace issue witht the S&B cases, but will experiment with this and other cases. I use a lot of Buffalo Arms cases in other calibers. They do a good job, but haven't tried in this caliber. I have both jacketed and cast bullets for the rifle. The jacketed 193 grs from Buffalo and same weight cast from Huntington. Factory ammo I have found from S&B and RWS. The RWS is very nice, but much more powerful than the S&B.
If you are working on using the 16ga barrels with Ball, check the diameter of the barrel at the chokes. Most are undersized from US standards. Of the three 16ga barrels on drilling or combo guns, the choke diameter is from .61 to .64. Your ball should just slip through the choke dimension. Look through the posts from DARYL on this forum and find out how he uses inverted gas seals to center the undersize ball in the barrel. There is good results with this method.
My drilling is much like yours and in the 9.3X72R / 16ga 2 1/2". The combos are in 9.5x56R x 16ga and the other in 12.7X44RX16ga. I just had wonderful cases made for the 9.5x56R from Rocky Mountain Cartridge co.
Here are a few photos of mine.