I believe that damascus steel barrels came in to common use with the industrial age because the steel was first rolled out in to ribbons, and then the ribbons were wound around a mandrill to create the barrel. The edges were heat welded. Later, a second and sometimes a third layer of steel was wound around the first so that seams were both welded and overlapping. That would probably be after the middle of the 19th century, however, when breechloading guns became common. By the time of the turn of the 20th century, and the appearance of smokeless powder, and " Nitro " powders, cold roll, and then hot roll steel was being produced, was cheaper to make, and then became even stronger than the older, more expensive damascus barrel process, and by WWI, such barrels were only available on special order. When the Great Depression hit, even those small shops that still offered Damascus steel barrels went out of business, and it was not revived after WWII.
I would think a single barrel fowler with damascus steel barrel would date to the 1840s, unless it was made for royalty in one of the arab countries, or in Japan. Damascus steel was made for hundreds of years and used to make edged weapons of all kinds. Its called " Toledo Steel", but actually was being made in the middle East, in India, and in China and Japan for hundreds of years before it came to Spain.