Hi,
With Backslpashes first example, I have to concede that Ketland could be the contractor however, he did not make the barrel. While I have seen guns with William Ketland marked locks and with private Tower proofs the only other barrel I've seen with WK is shown in Joe Puleo's paper on the Ketlands. William Ketland was a wholesaler, retailer, and exporter and he had his own business during 1801-1804. He died in 1804 but the business went on for several decades run by his partners. Your barrel probably dates from the very beginning of the 19th century.
dave