What is the significance of the eagle which Herod had put up over the door of the temple?

Simply put, the Golden Eagle was a symbol of Roman power and authority. Raised above the entrance to the Temple, it would represent the submission of Israel, and even the Temple itself, to Rome.

The OP also asks "Why did Herod put up the eagle?" It asks for "historical arguments based in academically credible source material." The thesis I will discuss, which is not one I am completely convinced of, argues that Herod did NOT put up this supposed Eagle. Its significance is therefore to denigrate Herod's memory and reinforce a long-standing tradition of Jewish martyrdom associated with heathen idolatry at the Temple.

Writing in the Journal of Jewish Studies, Prof. Jonathan Bourgel argues that "a careful reading of this account raises questions serious enough to reconsider its historicity... The episode of the golden eagle is a martyrdom narrative conveying a legend." Bourgel points out that the story of the Eagle is found nowhere outside of Josephus. He shows convincingly that, despite his ruthless character, Herod was careful not to offend Jewish sensitivities unnecessarily. He refrained from entering the Court of the Priests, for example. Yet, the heroic young men who tore down the Eagle would have needed to enter that area in order to perform their task, and it would be highly unlikely that they were all priests. Moreover their deed, as described in Josephus, would be physically implausible.

Regarding Herod's attitude toward the Temple, Bourgel says:

Josephus’ writings allude to Herod [elsewhere] as a man extremely respectful of the integrity and sanctity of the Jewish sanctuary. Thus, in the account of his capture of Jerusalem from Antigonus, Herod is reported to have used every effort, including wealthy presents, threats and coercion, to prevent his Roman allies from entering the Temple and thus profaning it. Likewise, he appears to have acted with great caution when he undertook the rebuilding of the Temple... He also made sure that only priests participated in the construction of the Temple itself, as entry to non-priests was strictly forbidden.

He concludes:

The episode of the golden eagle is a martyrdom narrative of legendary character, although it may preserve to a certain extent the memory of authentic tensions at the end of Herod’s reign... The account of the golden eagle affair can be likened to further legendary tales that contributed to the blackening of the memory of Herod after his death.

This is not to say that Josephus was lying about the Eagle. He was once a Jewish patriot to participated in the Great Revolt against Rome. As such, he may have known the legend of Herod's Eagle and believed it personally, even though it was supposedly been torn down decades before Josephus was born. No doubt many such stories circulated about the Herodian dynasty in the runup to the anti-Roman Revolt.

Conclusion: The Golden Eagle in Josephus' account is a symbol of Roman authority over the Temple. However, there may be reason to doubt the historicity of this story, which is not biblical and exists nowhere else in the historical record. Be that as it may, Herod's Golden Eagle serves to reinforce his reputation as a historical villain for Jews and Christians alike.

What is the significance of the eagle which Herod had put up over the door of the temple?

Herod was a pawn in the hands of Rome and the eagle signified the Roman authority in the Palestine region. The eagle was the symbol of the Roman army.

On this Herod showed his true colours and the populace revolted because of it.

To curry favor with Rome, King Herod put a golden eagle outside of the Temple of Jerusalem. Like other military standards, this eagle was carried into battle. The presence of the eagle meant the presence of the Roman Legion. By placing one at the gates, Herod was making a powerful statement regarding Jerusalem’s sovereignty under Rome.

Making idols was forbidden to the Jewish people, even if there was no attempt to worship them. Yet the Romans regarded eagle standards as holy symbols, anointing them on special days. Two respected teachers of the law, Judas and Matthias, spoke to religious scholars about this violation.

A group of these men pulled down the golden eagle and cut it into pieces. The king’s captain detained 40 participants, along with Matthias and Judas, and brought them before Herod. They explained that they made the choice to destroy the idol because they upheld the laws of Moses, and loved their religion.

Offended, Herod gathered Jewish leaders together in a theater. He lectured them about all of the things he had done for them, including improvements to their Temple. The Jewish leaders were all afraid of Herod, and relented. Herod had those responsible for the sedition burned alive. He appointed a new high priest: one subordinate to Herod and Rome.

The golden eagle in Jerusalem: History repeats itself