This is fascinating. I would love to have photos of your cane. I wonder how it got all the way to Scotland! Carmichael is not a name that rings any bells.
The 1800 date is a bit of a puzzler, though. Frederick Thatcher was born about 1826/7 in Chewton Mendip, Somerset s/o George & Mary. He had younger siblings Eliza, George and William. He moved to Abersychan sometime 1841–1851. He married Sarah Griffiths in 1858 and died in 1872. They had no children. The first mention I have found of Frederick Thatcher at the Full Moon is Slater’s Directory in 1859. I can find no record of the Full Moon before this date, either. His widow Sarah died in 1911. The Full Moon is last mentioned in 1914 and there is now a car park in its place. Like many Beer Sellers/Publicans Frederick had other employment. In 1861 as engineer in forge and in 1871 as engine driver (possibly an engine at a mine or works).