So why does the West, based as it is upon its Christian tradition, take sides in the ongoing dispute between people of the Book who reject Christ as Savior, privileging Israel above the Muslim world?

First, I would say there is a historical reason.  True, Christianity and Islam both have a militant history.  But Islam being the newer religion relatively, set out in a military way to expand their base, and in most cases it was at the expense of Christianity.  Worth pointing out that once they conquered a coountry, they often allowed peaceful coexistence.  But still they were on the attack and went deep into Europe at one point.
Second, another historical point, Christians fought back in the Crusades.  And of course also repelled the attack into Europe.  So there is a history of war between the two.
Third, I believe in a perception sense the extremists in Islam have coopted the religion.  Jihad is being used to attack the West,,,,at least that is the way it is being perceived by most Americans.  More mainsteam Muslims either have not stood up publicly, and more importantly when they do, MSM ignores them because it doesn't  play into the story MSM wants to tell--which sells a lot more newspapers and magazines.
Fourth, Islam now represents a very different culture than the West, which is largely Christian (or not religious).  So this is as much a cultural war, moreso, a cultural war than a religious one.  In fact, from the militant Islam viewpoint I think they are attacking the secularism of the West, and not really attacking Christianity.