Judea was a client state. But client states paid tribute to Rome, whilst raising taxes under their own authority. This proves the point, made in Luke, that under Herod the Great the form of enrolment for taxation was not Roman, but Jewish!
Now you are attempting to change what it says in Luke. He said the census was ordered by Quirinius. You must remember by now. He did not take over until 6 CE.
Subduction Zone Is right, you are trying to rewrite what Luke 2:1 say.
Luke 2:1 clearly stated that the order was by Augustus when Quirinius was governor of Syria, BUT neither Quirinius‘ governorship, nor his census occur in Judaea until in 6 CE, as Josephus stated 10 years after Herod the Great.
6 CE was also the time Augustus had Archelaus removed from Judaea, annexing the trierarchy as a Roman province. That's when the Roman census took place in Judaea.
No census were carry out when Herod was still alive, and for you to claim that Herod ordered the census, only make the gospel author either wrong or lying.
And in the 6 years of Herod's life, AND IF Jesus was born around this time, Gaius Sentius Saturninus (9 - 7/6 BCE) & Publius Quinctilius Varus (7/6 - 4 BCE) were the ones serving as governors of Syria, not Publius Sulpicius Quirinius (6 - 12 CE).
Josephus mentioned both Saturninus & Varus, and Varus was governor at the time of Herod's death in 4 BCE.
Clearly the author of gospel of Luke didn't know Judaea's history so well at this period, because he mixed the census with Herod's life, and don't understand Roman political systems.
The Romans wouldn't have census on any allied kingdoms or client kingdoms. It is only when they become new provinces that census would be used for taxation purpose.
And since Joseph appeared to be living in Nazareth, not in Bethlehem, according to Luke 2, as a resident of Galilee, Joseph wouldn't need to go to Bethlehem since he owned no property in Bethlehem.
Plus, the Romans wouldn't know about Joseph's ancestral tie to Bethlehem, which make the story in the gospel about traveling from Nazareth to Bethlehem, pointless.
Edit:
If Christians believed that Christian authors of gospels were divinely inspired by the Holy Spirit, do you think the author of gospel of Luke would make less mistakes and show less inconsistencies with Judaea and Roman history? Because of these mistakes clearly would show that the Holy Spirit isn't so infallible and inerrant.