"Bey Hardcastle is sometimes mentioned as being William James Hardcastle Bey. I am confused as to why."
"Bey" is a title, not a name.
"As with most Turkish titles, it follows the name rather than precedes it as in western languages, e.g. "Ahmet Bey" for "Mr. Ahmet". When one speaks of Mr. Ahmet, the title has to be written with a capital (Ahmet Bey), but when one addresses him directly it is simply written without capital (Ahmet bey). Bey may combine with efendi to give a common form of address, to which the possessive suffix -(i)m is usually added: beyefendim, efendim."
So whoever referred to him as "Bey Hardcastle" was incorrectly adopting western word order.