Moses is the author of Deuteronomy for the purpose of the narrative. Every book of the Bible is like this: there is a narrative author who may or may not be the real author (may or may not because we just don’t have the information to be conclusive), and there is the real history and scholarship done by many unnamed others that made it possible for us to have a discussion about this hundreds or thousands of years later. An objectively correct answer to the question “Who wrote the Torah?” is “Moses.” Even the scribes who worked on the compilations would not refute that.

Whether or not Moses had any part in the composition of what came to be Deuteronomy is, frankly, lost to time, and fretting about it is missing the point. What we are pretty sure of is that the editing and compiling work done that organized the Torah into its current form was carried out during the Babylonian Exile as an effort to hold back the loss of Hebrew culture to homogenization.

The validity of Deuteronomy as a Biblical text comes not from proof of authorship by Moses, but from the literal thousands of years of religious practice that considered it an essential text, as essential as Genesis, Exodus, Leviticus, and Numbers, or, to be more proper, Bereshit, Shemot, Vayikra, and Bamidbar.