JWs and many others make the common mistake of reading Luke as a separate work from the Acts of the Apostles, but they are actually one book (separated only as presented in the canon). So some of the things Luke writes in his gospel are actually dramatic foreshadows of what he says later in Acts. The “obedience” of Joseph and Mary to this "census" is supposed to stand in contrast to the revolt of Judas the Galilean who opposed it. (See Acts 5:37) This helps explain what Luke was doing with his “Infancy Narrative,” as it is called in critical scriptural study (something Jehovah’s Witnesses abhor like the plague).
While some of what is described may have a basis in fact, the individual details are purposefully woven together in such an anachronistic tapestry that they cannot, nor are they meant to be historically reconciled. Even the official Catholic Bible of the United States (the NABRE) mentions about this detail in a footnote to Luke 2:1:
Luke may simply be combining Jesus’ birth in Bethlehem with his vague recollection of a census under Quirinius...to underline the significance of this birth for the whole Roman world: through this child born in Bethlehem peace and salvation come to the empire.--Italics added.
Luke was as creative a writer as Cecil B. DeMille was a filmmaker (who, by the way, also retold Bible narrative with as much poetic liberty). There are two key factors which Jehovah’s Witnesses never point out about the Infancy Narrative of Luke which are commonly understood even by average members of Christendom:
It’s written in a different format meant to read like Jesus’ birth was no lesser than the great legends of the Hebrews. Luke 1:5-2:52 is written in a style that mimics the older Greek style of the Septuagint (some scholar believe it is actually not Koine). This peculiar style abruptly starts at 1:5 and in the same fashion ends just as suddenly with the first verse of chapter 3. If Luke were a filmmaker, this would be the equivalent of filming this section in black-and-white. He also adds “canticles,” or chanted prayers, much like adding musical numbers to a movie, in this section. (Note the "musical numbers" of: the Magnificat at Luke 1:46-56, the Benedictus at Luke 1:68-79, and even a third “song,” the Nunc dimittis of Luke 2:29–32.) Immediately this is a signal that this is not a direct report of history but an attempt to make the birth of Jesus take the form of a story or liturgical play.
There are peculiar comparative parables within that teach lessons within lessons. Luke writes things that cannot make sense as literal history, but they do as religious parables (an earmark of Jesus’ teaching style). Note that both Zechariah and Mary receive similar messages from Gabriel, but Zechariah is “punished” with speechlessness for asking "why," while Mary is portrayed as faithful for asking the same question. (Compare Luke 1:18, 20 with Luke 1:34-38.) Mary’s “song” precedes that of Zechariah’s, even though John is conceived and born first. Mary’s unborn infant even “baptizes” John in the Holy Spirit before he is born (and before John baptizes Jesus in the Jordan with water). Luke is demonstrating through an illustration that Jesus is John’s superior via this narrative device common to Hebrew legend-telling. (Note something similar in the infancy narrative of Jacob and Esau at Genesis 25:21-26.)
The Gospels are not meant to be journalism but religious retellings. The Jehovah’s Witnesses on the other hand crush even the religious life out of them by demanding that they be read as the equivalent of a journalist’s news report on the events. Matthew on the other hand is retelling the birth in the light of Jesus being the greater Moses: Jewish tradition holds that a star foretold Moses’ birth and that this was reported to the Pharaoh of Egypt which caused him to order the death of the Hebrew children; that Jesus’ birth is accompanied by similar events in reference to the Magi and Herod with Jesus fleeing to Egypt, Christ’s return from Egypt marks him in Matthew as the Greater Moses as Jesus return and literally enters the Promised Land whereas Moses did not after he left Egypt.--Matthew chapter 2.
The reason people don’t see these things when they are Jehovah’s Witnesses when you can open any commentary in Christendom or read these details in the footnotes of their very Bibles is indeed astonishing. I mean, you can walk into any bookstore or even go on line and read that the rest of Christianity knows these aren't meant to be taken literally. So obviously it shows how content the Witnesses are in being told they are right and how lazy they are in an attempt to double-check on themselves!