I researched this in my commentaries, most notably two respected Catholic Commentaries that I have, the Jerome Biblical Commentary and Catholic Commentary on Holy Scripture (note Jerome Biblical Commentary did not comment on this).
In both cases, the historical inaccuracy challenge is cited - both indicating that the historical facts given are somewhat problematic.
Collegeville’s in the most succint. This is a touchy area that many will dispute, but the Bible is inerrant in that which God intends to convey through inspired writing; most people equate that to faith, morals, and salvation plans. The Bible is NOT inerrant in science and history. This has been very clearly stated in council documents that have been cited on this site if you search. I can pull them out if you need. In short, an interpretation of the Bible where every word is considered inerrant is a fundamentalist view not supported by the Catholic Church.
So what did Collegeville say?
The census under Quirinius has caused much debate. Quirinius did not become governor in Syria until a.d. 6; shortly afterward he conducted the census in Judea that provoked the rebellion of Judas the Galilean (Acts 5:37). If Jesus’ birth is situated during the reign of Herod the Great (Luke 1:5), it cannot have taken place during this census several years later.
Among the various solutions proposed, the most satisfactory views Luke as associating a number of loosely related historical events around the time of Jesus’ birth in order to fix its context in the minds of his readers, without intending ironclad accuracy. The precise dating of these remote Judean events cannot have been too important to Greeks of the Empire seventy or eighty years later. But the realization that Palestine was part of the Syrian province at the time of Jesus’ birth might bring those events closer to Luke’s readers in Antioch, the center of the church’s missionary thrust in his time.
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I do not have an electronic version of Catholic Commentary, but it builds the case why there is historical precedent and motive for a census in Syria at that time, and it also says "It has long been admitted by responsible scholars that Quirinius twice held authority in Syria, the first time in conjunction with Sentius Saturninus, 9-6 BC, and there is documentary evidence in favor of this (cf. Lagr. S. Luc 65-6). It goes on with a much longer explanation of the time to do the census vs. the rules of Quirinius and Herod). It concludes by saying "In conclusion it may be said that a writer of proved carefulness like Luke would not have been guilty of such an error as could easily been detected by those for whom he wrote.
So two different theories, underlying the fact that the Bible is not a History book but must explain events of time as necessary for the reader ot understand God’s purpose.
BTW Catholic Commentary is very large and excellent, you can usually find it sold used at places like
abebooks.com since it is out of print. Jimmy Akin uses it all the time and mentions it on CA Live.