After much deliberation with Merriam Webster my quest to find the correct pronunciation of "Archmage" is still inconclusive. Detrimental to the cause is the apparent fact that Archmage is, in fact, not a word in the English language (at least, not according to Merry). So, I hoped that by anolyzing the pronunciation of the prefix "arch-" in various words I would be able to apply a rule even to a made up word. What I found was that when arch- is followed by a consonant it is pronounced "arch" (as in archbishop or archduke), while when followed by a vowel it is pronounced "ark" (as in archive or archaic). This would lead me to believe that since an "m" follows the arch, Archmage would be prounounced as spelled. However, "archenemy" does not follow this rule. Additionally, I cannot rule out "e" as an exception to the rule, due to words like "archetype" and "archeology." As such, though Archmage would appear to be pronounced in a certain manner, there is nothing to prove that it isn't an exception to the rule like "archenemy."
However, on dictionary.com, the French cognate "archimage" does register, but dictionary.com says it is prounounced "arkh-ih-mage" while google translate says it is pronounced "arsch-ee-mazh" (I know, [censored] google translate, but the French language is not in my repertoire (little joke there) and so I have no better resource, correct me if you can please).
Due to the fact that it is not technically a word in the English language and so cannot be declared to follow the apparent rule or be an anomaly, plus the confusion over how the French cognate is pronounced in English, plus the fact that Bethesda hasn't even made up their mind over the course of two games, I hereby declare the word "Archmage" to be pronounced "Arkh-meyj." Rebuttals?