I think there is general agreement that percussion guns became more prevalent by the second half of the 1830s. The original post asked about BD shotguns during the Mountain man era, which began much earlier, and was tied to the trade in Beaver pelts for the Fur Hat industry. Beaver hats were popular in both N. America, and in Europe until about 1830, when silk hats began to make inroads into the sales. By 1840, Silk hats had driven prices down on Beaver pelts so much that it was no longer profitable for the fur companies to send wagons out of St. Louis to rendezvous sites so that they could buy the pelts, and re-supply the trappers. Some sources indicate that the Mountain man era lasted from 1807 to 1840, some much shorter, a 15 year period from 1825 to 1840. Others talk about " mountain men " generally from 1820 to 1860. However, its clear that by the late 1840s, much of the plains had been settled, and the extermination of the buffalo was underway, as wagon trains forced routes from Missouri to California and Oregon.
Silk hats came into fashion in Europe in 1825, but became fashionable the world over by 1840, replacing the beaver hats.
So, to a real extent, the era of the Mountain Men ended just about the time that reliable percussion firearms, including DB shotguns were coming on the scene. Flintlock rifles and shotguns continued to be made after 1840-some argue with authority that the building of flintlocks has never ceased-- but industrialization allowed the manufacture of less expensive but reliable percussion guns, and they took over the majority of the market for guns. With industrialization, we also got cheaper import guns from Belgium and other European countries. America also benefitted by the immigration of many gunsmiths from Europe at this time, who decided to follow the market for their guns, and pursue their own fortunes. The explosion in development of firearms technology was possible in the 1850s, and 1860s largely because of this huge increase in our skilled work force. Where Flintlocks dominated firearms technology for 200 years, arguably percussion locks lasted only a mere 40 years at most. In the history of the development of firearms, from match locks, to wheel locks, flint, and percussion, before breechloading, self- contained cartridge guns, no system lasted a shorter amount of time than did the percussion system. The BB cap was available as early as 1846, and possibly 1842. Pinfire cartridges were available in the 1820. The first rimfire cartridge was made by Smith&Wesson in 1857, and has been continuously manufactured to this date. We know it now as the .22 short cartridge.
The end of the Mountain man era saw these men employed not as trappers, but as guides for wagon trains, scouts for the army, and, briefly, as Buffalo hunters. Compared to the period from 1807 to 1840, the life and fortunes of these later " mountain men" was quite different. Perhaps the most noble function these men served was to act as translators for the government and Indians, and as ambassadors to try to explain to Indians the terrible changes that their way of life was going to experience with the coming of European/American settlers. Jim Bridger was married to an Indian Princess, who bore him a son. His counsel was welcome by his father in law. Other trappers, like Jim Beckwourth , married and lived for years with Indian relatives, serving both as explorers, and guides, and translators for dealing with White Americans.