I recall discussing a much easier (but still very thorny) version of this problem in a class. Ultimately it came down to figuring out whether a particular god was Liar or Random. In desperation, one of the students suggested "Ask him
Are you a god?" On the grounds that a god who answers truthfully even some of the time, would never humiliate himself by answering "No".
I forgot the precise phrasing of that particular version.