London proofs. They should mean that the maker was a member of the Gunmaker's Company but by the time your gun was made, the company proof house (which was and still is privately owned) was allowing non-members to use their facilities. The mark in the middle, between the two London marks, is a maker's mark... a completely conventional marking.
Also by this time, many of the London "makers" were buying in semi-finished guns from B'ham, submitting the barrels for proof in London and finishing them in London in order to maintain the fiction of "London made."
There were two London members of the gunmakers company named Daniel Moore. The earlier one died in 1746, which is much too early for your gun. The latter of the two died in 1802 so if he was active to the end of his life, he could be your man. Its also possible that the mark continued in use after his death by other members of the family - this happened with the Wilson's - two or three generations of makers using the same mark. This is from Howard Blackmore's "London Gunmakers"... an essential reference if you're interested in English guns.
In looking at your pictures again, I think it is possible that the barrel is older than the rest of the gun... perhaps re-used from an earlier arm.