No it's not.

The mythology one is pronounced Karen

The moon is named partly after his wife so it sounds like Sharon.

nobody really speaks ancient greek.

I think that's something that a lot of people (astronomers, in particular) seem to forget. This would be like walking around town speaking Middle English. Nobody would understand you. You've never heard one of Shakespear's plays in the same langauge Shakespear did- he used Early Modern English, and every play has been adapted to modern audiences;

some more fun facts... Charon, the moon, was technically named after Jame's Chrsity's wife. (he's the guy who found Charon.).. her name was Charlene. though he did see the connection to the ferryman of greek myths and find that appropriate as well. he pronounced it Shar-on

So. All that to say... Pronounce it how you like, there's options.