I'm afraid I can't really address the Jewish side, except agreeing that the Jewish people under Roman rule in Palestine did have a, shall we say, rocky history with their rulers.

But as for the texts of the New Testament, I would disagree that there was such a strong need for reconciling them with Roman government. In the Gospel narratives, Pilate is portrayed as reluctant to condemn Jesus; at worst, his decision can be chalked up to weakness or political expediency, but the evangelists go out of their way to move the blame away from Pilate, highlighting his desire to release Jesus, and onto the Jewish leadership in Jerusalem, the priests and Pharisees and Temple leadership.

Other NT texts also seem favorable towards Rome, or at least favorable towards peaceful coexistence and life under their rule. Romans 13 and 1 Peter 2 speak about submitting to the authorities, paying taxes, and honoring the emperor.

Writers in the early Church received this attitude from the NT texts; Tertullian, writing in the 3rd century before Christianity's legalization, noted the text and encouraged obedience. His advice was to obey the authorities whenever possible, so long as it did not require idolatry or disobedience to God.

Not that the early Church's relationship with Rome was always pleasant and trouble-free, of course; there are certainly strains of scholarship that seek to uncover hidden anti-Roman sentiment within the New Testament, and see the superficial reading of the text as a self-preservative smokescreen to keep the Romans off their back. But as the Imperial government over the course of the 4th century went from persecuting to endorsing Christianity, the texts of the New Testament were no insurmountable obstacle.