R: ”It is very important to me to confirm that weapons must have actually existed around 1872. Can you make any assumptions about its probable period of production?”
Yes, there was a huge national roundup of weapons conducted, by prefecture, in 1872 ... which was the year of Jinshin 壬申 (as it says on your dai-kabu). This means that the gun is older than that.
Your gun does not have many typical geographical or ryu-ha gun school characteristics, making it more of a generic weapon for light military use on a battlefiled anywhere, a typical Tanegashima-style gun produced throughout the Edo Period. I am guessing that the zogan barrel decorations were added at a later period to increase its sale value, probably Meiji or later when plenty of artisans were still alive and available to do the work. Use of matchlocks began to die out around the 1840s and 1850s as they were being gradually replaced by influxes of versions of Western guns.
Without a Mei and without closer study I would hesitate to narrow that down any further than 'Edo Period, likely any time between 1600-1840'.
(Three things do hint to me that it could be earlier within that period, i.e. the wide 'za' pin surrounds, the narrow-waisted pan, and the square shape of the sights.)