If the Holy Spirit inspired the whole Bible, why don't
the books all possess the same style of writing? If God never changes, why would his writing styles change so radically? Doesn't this prove the Holy Spirit wasn't the "principle author" of each book in Scripture?
No, it proves that the Holy Spirit, although the principle author of each book in Scripture, worked through human authors, preserving and making use of each one's particular style of writing. Catholic theologians and Scripture scholars in the early Church used a particularly apt musical analogy. They explained that, when a piece of music is played on various instruments, it will obtain a different sound and aural texture from each one, yet each rendition will be the same melody coming from the hand of the same composer.
The Star Spangled Banner, when played on a harmonica, piano, clarinet, guitar, tuba, or a kazoo, will sound markedly distinct on each different instrument, yet it's the same song being played in each rendition.
The same is true for the books of the Bible. The Holy Spirit, like a composer, selected different men to be the inspired "instruments" through which the melody of Scripture would be "played." That's why the style and elegance of the Greek composition of Luke's Gospel contrasts with the terse style found in Mark's Gospel, and the Old Testament books differ widely in their choice of vocabulary and literary style.
In each case the Holy Spirit inspired the biblical writer to write what he wanted written, all the while preserving, in an admittedly mysterious way, their free will and personal style of expression. To learn more, read the encyclicals Providentissimus Deus (Leo XIII, 1893), Spiritus Paraclitus (Benedict XV, 1920), Divino Aflante Spiritu (Pius XII, 1943), and Humani Generis (Pius XII, 1950), and don't omit Vatican II's Dei Verbum.