Off-topic, but in France to “take French leave” is “filer à l’anglaise”.
I believe the Entymonline cite from the 1560s for the noun desparate is the obsolete sense of person without hope, which matches the OED’s earliest cite of 1563 for that sense. The OED’s earliest cite for desperado with that definition is 1610. The 1647 date is the earliest cite for desperado in the sense of outlaw.
There’s also the obsolete word “grenado” (which appear to be cod Spanish for “grenade”).
I don’t know what “cod Spanish” means, but the word for “grenade” is granada: for the city (there are many, btw), for the country (both capitalized, of course), for the fruit and for the hand thrown bomb. The word “grenado” is in no Spanish dictionary of mine.
“Cod,” as you have gathered, means “ fake, phony, mock ,” and thereby hangs our tale. “Cod” meaning “phony” has been British slang since the early 20th century,
Thank you, good link.