At a guess, in Zeithain the author simply makes a concession to modern sensibilities and assumes the more historical ignorant of his readers would be put off by two lovers calling each other by title and last name. (I think he's wrong; Stephen Frears's film version of Dangerous Liasons was a hit here, too, and the German dubbing correctly had everyone in the cast use "Sie" for each other, including post coital moments, with the only "du" being heard was Valmont adressing his servant. It was a smash hit over here, too.)

Is my idea plausible that 18th century Katte, far enough along in their relationship, might be invited to speak that way to Crown Prince Friedrich in private, intimate moments, but would still be "vous/Your Royal Highness"ing him in all other contexts, including the execution with 150 witnesses?

IMO yes, especially since by definition no one else would have been present in private, intimate moments to record it.