...While such logical inconsistency is certainly not limited to Islam, it opens up a larger question as well: if you believe part of your religion isn't true, should you modify the religion and keep following it or are you compelled to do away with the entire religion?

It is in our nature to have contradictory beliefs.  That is how we ground our faith in the truth of what we do  finally come to believe in--by arriving at the conclusion that an assertion's contradiction is less viable when we attempt to defend it against our experiences and render it consistent with other beliefs.  We always weigh opposing points of view, and they must be felt compatible with other things we believe about reality.  So there ought to be nothing wrong with abandoning parts of a religious doctrine if those parts become too burdensome to defend in our minds. 

As I see it, the dilemma you raise has more to do with the nature of religious faith in general.  It is often based on little more than trust in the authority of those who first indoctrinated you in the faith.  If you can question any part of it, that throws the entire authority in question.  You have to have some basis for rejecting earlier beliefs.  Christians were able to reject large parts of OT doctrine precisely because Christ represented a new authority.  Islam has its own schisms that find ways of discounting or reinterpreting the more virulent sections of scripture, just as Christians did.  Unfortunately, Muslims seem to have undergone considerable backsliding into fundamentalist doctrine since their earlier, more tolerant behavior.  The huge increase in oil wealth has done more than anything else to spread Wahhabist fundamentalist beliefs.  Hence, a more virulent brand of Islam has come to be seen as representative of its core beliefs.

In short, I'm looking for input on the merits of "moderate" Muslims ignoring/opposing some of Islam's teachings about violence with, if desired, some thoughts on the larger issues stemming from that (mentioned briefly above).

During its golden age, Islam seemed a far more tolerant religion that Christianity during the same period.  Jews, for example, tended to fare better in Muslim lands than in Christendom.  (Consider, for example, the situation in Moorish Spain.)  Nowadays, the situation seems rather reversed, as Christian countries have evolved more secular governments to reduce the effects of inter-faith squabbling.  The Muslim world has some secular governments, but even they are under strong attack (e.g. modern Turkey).