You are assuming two things here...
1. That the census referred to in Luke was the census under Quirinius of 6–7 A.D. It is not, since the census mentions the entire Roman world took part, not just Syria and Iudaea.
2. That the bible must have an error in its chronological narrative, it does not, Jesus was in fact born under Herod the Great, who contrary to popular perception died in 1 B.C. not 4 B.C., this is now admitted by a number of historians.
As I said earlier, the census of Luke was not a census at all and is actually referred to by Josephus himself...
For there was a certain sect of men that were Jews, who valued themselves highly upon the exact skill they had in the law of their fathers, and made men believe they were highly favored by God, by whom this set of women were inveigled. These are those that are called the sect of the Pharisees, who were in a capacity of greatly opposing kings. A cunning sect they were, and soon elevated to a pitch of open fighting and doing mischief. Accordingly, when all the people of the Jews gave assurance of their good-will to Caesar, and to the king's government, these very men did not swear, being above six thousand; and when the king imposed a fine upon them, Pheroras's wife paid their fine for them. In order to requite which kindness of hers, since they were believed to have the foreknowledge of things to come by Divine inspiration, they foretold how God had decreed that Herod's government should cease, and his posterity should be deprived of it; but that the kingdom should come to her and Pheroras, and to their children. These predictions were not concealed from Salome, but were told the king; as also how they had perverted some persons about the palace itself; so the king slew such of the Pharisees as were principally accused, and Bagoas the eunuch, and one Carus, who exceeded all men of that time in comeliness, and one that was his catamite. He slew also all those of his own family who had consented to what the Pharisees foretold; and for Bagoas, he had been puffed up by them, as though he should be named the father and the benefactor of him who, by the prediction, was foretold to be their appointed king; for that this king would have all things in his power, and would enable Bagoas to marry, and to have children of his own body begotten.
The reference is that all the people were required to offer an oath of allegiance to Augustus Caesar, this was done throughout the Roman world, and also in Israel, since Herod was known as friend and Ally to Caesar.
Documented archaeological evidence of this is also found in Paphlagonia (north central Asia Minor) that is clearly dated to 3 B.C. records an oath of obedience "taken by the inhabitants of Paphlagonia and the Roman businessmen dwelling among them." (Lewis and Reinhold, Roman Civilization, II.Pages 34–35).
In the third year from the twelfth consulship of the Emperor Caesar Augustus, son of a god, March 6, in the … at Gangra, the following Oath was taken by the inhabitants of Paphlagonia and the Roman businessmen dwelling among them:
“I swear by Jupiter, Earth, Sun, by all the gods and goddesses, and by Augustus himself, that I will be loyal to Caesar Augustus and to his children and descendants all my life in word, in deed, and in thought, regarding as friends whomever they so regard, and considering as enemies whomever they so adjudge; that in defense of their interests I will spare neither body, soul, life, not children, but will in every way undergo every danger in defense of their interests; that whenever I perceive or hear anything being said or planned or done against them I will lodge information about this and will be an enemy to whoever says or plans or does any such thing; and that whomever they adjudge to be enemies I will by land and sea, with weapons and sword, pursue and punish. But if I do anything contrary to this oath, or not in conformity with what I swore, I myself call down upon myself, my body, my soul, my life, my children, and all my family and property, utter ruin and utter destruction unto all my issue and all my descendants, and may neither earth nor sea receive the bodies of my family or my descendants, or yield fruits to them.”
The same Oath was sworn by all the people in the land at altars of Augustus in the temples of Augustus in the various districts. In this manner did the people of Phazimon, who inhabit the city now called Neapolis, all together swear the Oath in the temple of Augustus at the altar of Augustus.
This is the only "census" that included the entire Roman world. The reason for the misunderstanding is simple, the actual Greek says "enrollment" or "registered" but it is translated "census" or "taxed" into English.
Used in a common phrase, the meaning takes on a different idea... I enrolled for a course in history, for this semester.
The word is used also in Hebrews 12:23.
23 to the church of the firstborn, whose names are written in heaven. You have come to God, the Judge of all, to the spirits of the righteous made perfect, (NIV)
23 and to the assembly of the firstborn who are enrolled in heaven, and to God, the judge of all, and to the spirits of the righteous made perfect, (ESV)
And yet it is the very same word, ἀπογραφή apographē.
Sorry did mean to say earlier