Under what genre is the Bible classified?
Certainly, the bulk of the Bible is either narrative or poetry. You've left out, however, one very important element of the Hebrew Bible, namely, LAW.
The books of Exodus, Leviticus, Numbers, and Deuteronomy contain long passages of the laws that God gave to Moses on Mt. Sinai. Leviticus, in particular, contains very little narrative at all, being almost entirely in the form of a law code.
The Biblical law codes cover civil law, marital law, monetary law, laws of religious rituals, laws of animal sacrifice, laws of ritual purity, and many other fields. Jewish tradition calculates that there are a total of 613 statutes in the Five Books of Moses.
It is true that the laws are presented in the context of the historical narrative of the Hebrews leaving Egypt and then receiving the Torah (the Law) at Mt. Sinai. It would be hard to argue, though, that the author (or Author) of the Bible intended only to tell the story about how a law code was once upon a time given to the Israelite nation. Numerous verses indicate that the author considered these laws to be binding for all generations.
Thus, we must add LAW or LAW CODE to the list of genres that are used in the Bible.
The whole book is considered to be a historical narrative.
Contained within the Old and New testaments of the Bible are the broad genres of narrative and poetry.
If you break down the poetry into more specific genres, you will find lyric poetry, lament, love poem, epithalamion(wedding poem), and nature poem
If you break down the narrative genre into more specific genres you will find creation stories, legal codes, parables, letters, epic, tragic comedy, etc.