This reminds me again of a key reason why apologetics can be counterproductive, because their content appears to focus on elucidating the past, but it becomes clear that they only care about "defending the Bible".
Most readers aren't looking for dogmatic justifications so much as a rationally plausible scenario.
Therefore, if your true goal is defending God's Word, you might do best to admit ignorance on this one.
But if you're interested in the historical census of Luke's Gospel, we've discovered Quirinius is Irrelevant.
Whether anyone knows it or not...