I show a slew of 2" examples in the S867xxx and S868xxx range. One of them is David's gun, mentioned above. Most of them shipped in January and February of 1947, but I do show a couple fliers. One shipped in August, 1947, and the other in September of that same year. So January or February is a good guess for S867731, but it clearly could have shipped later. So much depended on demand and the 4" and 5" guns seem to have been outselling the 2" examples (the 6" as well) by quite a large margin.
The nearest number in my personal collection is one with a 4" barrel, serial number S861363. It shipped in December, 1946.
If you really have to know, you will need to purchase a letter.