You're probably getting a bit confused because Herod was from a Hebrew royal dynasty, but his successful ascent to the throne (after bloody internal power struggles in the region, and even within the same Herodic royal family) was a direct product of Rome aiding him and installing him militarily.

I'm not confused about this. It was a client kingdom, not a Roman province until 6 CE, which is when the Census of Quirinius (the one you mentioned Josephus recording) was done to reckon what taxes should be levied.

The Gospel of Luke uses the Census of Quirinius to date the birth of Jesus but also cites the time of Herod the Great (who died between 5 BCE and 1 CE), while the Gospel of Matthew only references Herod the Great. Most critical scholars agree that Luke is in error, while some religious scholars have attempted to defend the gospel, sometimes invoking unproven claims.

(Emphasis mine)

In order to install an ad valorem property tax in the new province, Publius Sulpicius Quirinius, the legate (governor) of the province of Roman Syria starting in 6 CE,[2] was assigned to carry out a census in Judaea.

Scholars point out that there was no single census of the entire Roman Empire under Augustus and the Romans did not directly tax client kingdoms; further, no Roman census required that people travel from their own homes to those of their ancestors.

Is it possible you've been misinformed about how historically accurate the bible is?