If the deuterocanonical books don’t claim divine inspiration, how can they be included in the Bible?

No book of the Bible claims itself to be divinely inspired. Divine inspiration means that God himself authored the exact words of the text (using the human writer’s mind, personality, and background), and no book states anything like, “The words of this book were chosen by God” or “This book is divinely inspired.”

The term “inspired” (Greek, theopneustos) only occurs once in the Bible (2 Tm 3:16), where we are told that all Scripture is inspired. We first know that something is Scripture and then infer that it is inspired; we do not first know that it is inspired and then conclude it is Scripture.

The only non-technical references to inspiration occur when one book of the Bible reports that God or the Spirit spoke through the words of a different book (for example, see Heb 3:7-11, concerning Ps 95). In no case does a book of the Bible state this for itself. Even if it does claim to contain divine revelations or visions (as does the book of Revelation), it does not say of itself that every word of its text was inspired. That is something we must infer from 2 Timothy 3:16. Since no protocanonical book of the Bible meets your test, it can scarcely be expected of the deuterocanonical books.

Claiming to be inspired is a different thing from really being inspired. The Book of Mormon claims to be the Word of God, but isn’t; the Gospel of John doesn’t, but it is. To determine inspiration, one must use an external authority for verification, and the Church is the only institution that can be that external authority.