His arguments are excellent and detailed, so I wanted to reproduce them in fairly full form, as that will help some people.
But it may help others to have a much more pithy summary of the main objections, and the replies you can give if someone asks you about this.
Objection 1: Luke is inaccurate
Objection: Luke is not historically reliable. He says Caesar Augustus ordered a census in the time of Quirinius. There wasn't one.
Answer: Luke takes great care to research his historical background (Luke 1:1-4). On every occasion when we have enough information to check something he mentions, it checks out. It's a big leap from saying we don't have a record of something outside the Bible, to saying that it never took place, especially when dealing with a Bible writer who is usually both careful and accurate. What exactly makes you say that Luke is definitely wrong here?
Objection 2: The Romans never took a census of the entire empire
Objection: Luke 2:1 mentions a census of "the entire Roman world". But the Romans only ever took censuses of at most a province at a time.
Answer: The Romans loved their censuses, because they loved collecting their taxes. During this period, we know of several provinces where the Romans were taking regular censuses (such as Syria, Gaul and Spain). It seems it was a regular custom to survey the empire, one region at a time. When Luke says that there was a census of the entire Roman world, he was probably referring to this regular custom, and not saying that a single list was ever made up of the whole empire.
Objection 3: No Roman census would require Joseph to go to Bethlehem
Objection: The Romans just counted people where they were
Answer: The Romans also respected local customs to avoid causing unrest. Jewish culture would have required people to be registered according to their ancestral land, which remained within the clan even if people moved away. So it's entirely plausible that this census would have respected this particular cultural marker.
Objection 4: Why did Mary have to go to Bethlehem?
Objection: If Mary was engaged to Joseph, she wouldn't have accompanied Joseph.
Answer: She may not have been required to go with Joseph, but the Romans may well have allowed her to go if Joseph requested it. She was heavily pregnant, and he wouldn't want to miss the birth. It's making a lot from silence to insist that we can be sure the Romans would refuse such a request.
She also may have been married to Joseph by this point. The word for "pledged to be married" in verse 5 could refer to an unconsummated marriage, and this would certainly fit with Matthew 1:25 which makes exactly this point. If that were so, she would have gone with him.
Objection 5: The Romans wouldn't have taken a Judean census during the time of Herod the Great
Objection: Herod ruled like a local king; he even minted his own coins. The Romans wouldn't have interfered in this way.
Answer: The Romans never took their eye off their empire. They may have delegated a lot to their vassal, Herod, but they would still have collected taxes. In fact, we know they did. Herod the Great's territory included Samaria, which went to Archelaus after his kingdom split in 3 when he died. The Romans immediately reduced the taxes for Samaria, which tells us that this part of Herod's kingdom was being taxed (and so censuses would be taken) during Herod's time.
Objection 6: This must have been the census from A.D. 6, referred to by Josephus
Objection: Josephus mentions a census in Quirinius' time, in A.D. 6, but of no censuses by Quirinius from before that. In fact, he mentions that this census was an innovation, and so caused a revolt, which suggests that it was the first of its kind.
Answer: Once again, be careful arguing from silence. Josephus may not have mentioned an earlier census, but his accounts are not exhaustive. There could still have been one. In fact, there were specific ways the census of A.D. 6 was innovative and so caused unrest. In particular, Rome was re-establishing their presence in the region, and Archelaus was seen as a traitor to support this. It was this, rather than the very fact of a census, that caused the problems. This would have made this particular census of interest to Josephus, whereas earlier ones may not have deserved a mention.
Objection 7: Quirinius was not governor of Syria at the time of Jesus' birth
Objection: Quirinius did not become governor of Syria until 6 A.D. Jesus was born in the time of Herod the Great, who died in 4 B.C. So the dates do not match.
Answer: There are gaps in our knowledge of the history of this period. Those gaps mean that we cannot give a definitive explanation of who ruled where, and when, such that we can say this accounts for Luke's phrasing. But there are a number of possible ways of explaining Luke's account given what we do know, and those gaps also mean that we cannot say for certain that all those possibilities are wrong and that Luke is therefore mistaken.
It may be that Quirinius was also governor of Syria from 4 B.C. to 1 B.C. That would still be after the death of Herod the Great, but would be close enough that Varus (his predecessor) could have ordered a census that was completed and written up once Quirinius had taken over, meaning his name got put to the final census.
Alternatively, it could be that Quirinius was enlisted as the administrator to organise the census, since a census of this scale would be a complicated task delegated to an able administrator. The word for "governor" here could mean "administrator" in this sense.
Alternatively, the word Luke uses to describe a census "while" Quirinius was governor of Syria may also be translated "before". In that case, Luke is explicitly saying that this census was not the well-known one from A.D. 6, but is a census from before that period.
Again, we do not know enough to say for sure which of those options is correct (or even, possibly, a different option again). But we also do not know enough to rule any of those out. Given this, it would be safer to trust that a careful historian like Luke was accurate here too, even if we (being much further from the events) cannot explain exactly how his account reconciles with other facts we do know.