Surely you are aware that Luke is not the only Gospel writer!?! Matthew also gets Jesus born in Bethlehem, but not through a census or taxes. Rather, Matthew has Joseph have a dream forewarning of Herod's intent, after meeting the Magi looking for where this new "King of the Jews" is to be born. Mark always refers to Jesus as being from Nazareth, and nwever once metnions Bethlehem. And Mark, written about 66 AD or later, is the earliest Gospel, at least 35 years after the death of Jesus.
John written at least 30 years (and likely more) after Mark, was not unaware of Bethlehem’s significance. John mentions a debate where some Jewish people referred to the prophecy which claimed that the messiah would be a descendant of David and come from Bethlehem. But Jesus according to John’s Gospel is never associated with Bethlehem, but with Galilee, and more specifically, Nazareth.
Finally, remember this: The Gospel of Mark probably dates from c. AD 66–70, Matthew and Luke around AD 85–90, and John AD 90–110. Despite the traditional ascriptions, all four are anonymous and most scholars agree that none was written by an eyewitness. So, really, why are you placing so much emphasis on Luke?