How would one substantiate that the LXX in Jesus' time lacked the Deuterocanon? The definition of the LXX is malleable. It is technically only the 5 books of Torah translated by the LXX 70 Scribes. But in commonly parlance by scholars it means the Greek version of the OT Bible used in ancient times. So if one says, "The LXX does not include the Deuterocanon because the LXX is only the books of the Bible," then one is using circular reasoning in defining the LXX. You would have to prove that the LXX in particular lacked the Deuterocanon, not just that the Hebrew version or what one considers to be the OT canon lacks the Deuterocanon.The version of the LXX that contained the Deuterocanon, which included the Greek additions to Daniel, wasn't until sometime after the first century AD. The version that Jesus & the apostles used did not include it, because the Deuterocanon was not laid up in the Temple, since the Sadducees who were the Jewish sect that was responsible for laying up books there did not accept the Deuterocanon. Also, according to John Martignoni from EWTN, the LXX was completed around 134 BC. Most (if not all) of the Deuterocanon was written or translated after this date. The later Masoretic Text compiled around AD 200 simply reflects the books that were in this BC version of the LXX, what was laid up in the Temple, & the version that Jesus & the apostles used, which all excluded the Deuterocanon.
Scholars commonly consider some Deuterocanon books as written before 150 BC. Further, the Greek version does have differences from the Hebrew one when it comes to the same passages, and often the Greek NT uses the Greek version instead of the Hebrew one. Further, Greek was a lingua franca in the Middle East and the apostles knew Hebrew, Aramaic, and Greek. Therefore, the fact that the Hebrew version was in the Temple does not disprove that Jesus and the apostles used the Greek version too.
Further, the NT repeatedly refers to ideas and passages specifically in the Deuterocanon. At the moment, I am listening to lectures on the books of the Deuterocanon, because I am not very familiar with them, and the narrator notes that the story of the widow with 7 dead husbands in Jesus' debate with the Sadducees is in Tobit.
Luther said that the Deuterocanon is useful reading, and I am learning new things that are alluded to in the NT.
The Muratorian Canon from c. 170-200 AD, one of the earliest Christian Bible lists, includes some of the Deuterocanon as being canonical.
If you were living in 50 AD or 200 AD in Italy, you would have a big mass of spiritual Jewish books in Greek, and they differed from the Hebrew OT used then in many of the same verses. The earliest Greek codexes as collections of Greek Bible books include the Deuterocanon. Further, the Greek version of Daniel has chapters called Deuterocanincal.
An early Christian Greek Bible translator, Origen, included some of the Deuterocanon in his translation:
Armin Lange writes in his “Canonical History of the Hebrew Bible” in the Textual History of the Bible Volume 1A (pp. 35–81),
Origen included at least Ben Sira, Wisdom of Solomon, and Judith into his Hexapla.