Yes, fug boots were Hawker's invention. I've no certainty about the source of the word, but I seem to recall "fug" being used to describe the warm, closed-in (read "odiferous") atmosphere of a typical RFC squadron mess. By extension, then, "fug" boots would be something that kept you warm.
I think you are right. What you need on a cold wet December night in England is a pub with a good 'fug' to it. If that is the derivation, Hawker would have been using the term somewhat ironically.
I've seen a real pair for sale on Ebay, price about $1000. Repos I've never seen but maybe a place like Great War would make you a pair? I seem to recall they will do things like that but it will still be pricey.
FUG - Brit. informal - a warm stuffy or smoky atmosphere in a room. ORIGIN late 19th century (originally dialect and schoolchildren's slang): of unknown origin.