I am not a native speaker, but I see a major difference between arch- in archenemy or in archaeology and even another one to archipelago, which would explain the different pronunciations.
In the first case it is used as a prefix. Enemy is still a word by itself, as bishop or diocese are. The arch- prefix is used to emphasize the relevance / importance / significance of this special enemy / diocese / bishop.
Here arch- derives from the Greek archi- or arkhi- (main, chief) as you mentioned before.
This is also correct for archipelago (archi meaning main), but here it is a word that can only be used as a whole. Pelago has no meaning in English. Archipelago derives from Italian (where it comes from Greek), so one cannot compare it with archbishop etc. It has not been combined from already English words.
In archaeology arch- is no prefix. Instead archaeo- could be counted as one. The word derives from the Greek words archaios (old, archaic) and logos (science).
The difference becomes clearly visible when translating to German:
Archbishop - Erzbischof
Archenemy - Erzfeind
Archdiocese - Erzdiözese
Archipelago - Archipel
Archaeology - Archäologie
As a prefix to the given words Erz- has the same meaning as arch- and like archipelago the German word Archipel derives from the Italian word. Archaeology and Archäologie are just the same.