This detail can be taken as a symbolic device that comments on the absurdity of the workings of human faith.
In a story that is building up to a massive fire-bombing and which includes mention also of atomic weapons, a small bullet-proof Bible seems to be a rather silly mode of protection. The usefulness of such an object in the face of the powerful forces of warfare in WWII is certainly questionable.
Many oddities and foibles are discussed in the novel, and the Bible here is just one example. These oddities of faith are often contrasted to a sense that humans lack control over their fates.
When we consider the idea that fate is already fixed - according to the Tralfamadorians and according to the novel's author who has already told us what will happen to all the characters very early in the book - the prospect of a pocket-sized piece of paper and metal acting as protection becomes patently ridiculous.
While assigning a definite meaning to the bullet-proof Bible, we can reasonably apply the above interpretations to it.