Everything I have ever heard from etymologists is that there is no clear or even fuzzy reason it is called a jew's harp. It just appears in various publications - sorry I don't happen to have the Oxford Enlish Dictionary with me, my girlfriend's got it - several hundred yeas ago.
The other terms - jaw and juice appear later and are considered "retro" derivations, basically people trying to explain away the term jew's. There is no evidence that it is a derogatory term. It is likely to be a coincidence of three letters. Possible a regional term that somehow became the accepted general term. If you don't think that can happen, check out a list of the words Shakespeare either invented or borrowed from friends around Stratford.
There is political correctness and then there is political correctness. The Jew's Harp Guild and presumably its members do not have a problem with the term. One ould hope the organization isn't anti-semetic. check out www.jewsharpguild.org