There is a 108xxx on the list that shipped in July 1922, so that confirms the ballpark.

While Ralph is right that these were catalogued until 1940, the general assumption is that production petered out at least a decade earlier; the caliber rapidly lost popularity in the 1920s, also a reason why shipping dates on higher serials can be hard to extrapolate with any reasonable accuracy.

To add a tidbit to Ralph's dissertation, by the time yours would have been sold, the "Model 1905" designation was no longer in use (S&W ended that by the end of WW I), and the numbered Changes were never used by anybody back when the guns were actually made, sold, and used. I don't generally use them unless there is a good reason, like when specific parts are the issue. But just like the screw count, which on the M&P usually serves no purpose either, you'll sound more learned and make some oldtimers here happy if you use them