Ralph, I have a 3rd model perfected (DA) Single Shot, but with a 2nd Model barrel. The Barrel S/N is 3844 on both flat and latch, and it has the 2 rear screw rear sight and single line address. But the barrel has a flat bottom to mate with the flat of the frame. The barrel bottom appears very original, with a beveled edge, and groove down the center of the barrel flat, It has been impossible for me to see the extractor number so far. The gun frame number is 8694. You mentioned the guns are an offshoot of the 38 Single action, but I have found the 3rd model is patterned after the .32 Hand Ejector. My gun is not nearly as pristine, and has been re-blued and when i obtained it I think was a put together. Someone had replaced the rebound spring slide with a filed down over-size one, (possibly a k frame one?). The side plate wouldn't screw down snug, and when I attempted to snug it down the action would bind. I replaced it with a .32 Hand Ejector slide and it has worked perfectly since. It does have the checkered 2 screw small gold medallion stocks, but I am unable to see any indication of a number penciled on the back of the right panel.
I am only mentioning all this because of the 2nd model barrel, with a flat bottom and S/N of 3844, within 163 of your # 4007. I can furnish pictures if wanted.
The one picture doesn't show the barrel bottom flat completely, but does show it is flat, and evidently originally that way.