Actually, they are Ordnance Private Proofs (no BPC and V in the scepters), done at the Ordnance proof house for the trade. the barrels were shipped to London for proof via the fantastic English inland canal system. This was so efficient that they were probably back in B'ham to be stocked in a fortnight. The practice probably stopped around 1805 but that is only a guess. The records are fragmentary but they stop in the 1804/05 period so we have no way of knowing if private proofs continued after that time. It definitely stopped by 1813 when the B'ham proof house opened. As for the date... the owner is all wet though that is understandable because the commonly published dates for the Ketlands place them in the export trade much to early. The founder of the company was only born in 1737... he was a teenager in the 1750s.
Shortly before the B'ham proof house opened, in 1813, there was a piece of legislation introduced in Parliament, sponsored by the London gun trade. It required that all barrel markings denote where the barrel was actually made - which would have ended the custom of putting "London" on B'ham made barrels. The legislation went absolutely nowhere when the B'ham makers pointed out that virtually all of the London makers were buying their barrels semi-finished from B'ham, finishing them and having them proved at the London proof house... so even the "real" London barrels were B'ham products.
Today we are surrounded by meaningless advertising hype, so much so that most of us barely notice it... it was exactly the same in the 18th and early 19th centuries. In fact, it was probably worse since no one had even thought of "truth in advertising".
As a finished gun, I very much doubt that this gun, and tens of thousands like it, ever saw London unless it was from the hold of a canal boat taking them down to the Thames for shipment to America. Just as many "London" guns were probably shipped from Bristol and Liverpool and never got closer than 150 miles to London. In fact, last Sunday I was in London and located the site of the Ketland warehouse. It is in easy walking distance of the Thames embankment in a neighborhood once thick with the offices and warehouses of commercial export firms.