No, it still fails no matter what. Even if Herod did not die in 4 BCE, I have seen dates as late as 1 BCE by an apologist and that was the furthest that he could push it we still know when the Census of Quirinius was and why it was held. Herod's son Archelaus took over the running of the major part of Judea, which was probably why it retained that name, did begin in 4 BCE. Romans record events like this. Maybe Herod was not dead. That would still mean that he was no longer king. By the time nine years had gone by Archelaus was such a failure as a ruler that rioting had begun and the Roman Empire felt that they had to step in and they did deposing him. Taking over the country as part of the Roman Empire. Which meant that they would have to pay Roman taxes. That was the reason for the Census. Quirinius, who had recently taken over Syria, organized a census. Something that could not be done in Judea/Israel under a Jewish ruler since it went against Jewish law. The Roman's did not give a rat's behind about Jewish law. But the census did cause a revolt. Again, Roman's record these things. Very traditional Jews opposed it. The leader of the revolt was Judas. No, no, no, not that Judas, like John and even Jesus that was a common name back then. Judas of Galilee, not Iscariot, began a revolt in 6 CE and that was even mentioned in Acts. And Josephus blames that as the start of the war that ultimately resulted in the destruction of the temple 60 years later. The dude even has his own Wikipedia page:
en.wikipedia.org
You will hear a bunch of lame attempts by apologists, not historians who say "Well maybe . . . " but they are all rather weak and no one pays them any serious attention. You know what would help them? Historical evidence, yet the apologists cannot seem to find any. This is another example of why one should not take certain parts of the Bible literally.
The story starts out well enough. It does not go into why Judea had to be taxed. As a client state under Herod their populace would not be taxed. The client states were somewhat subservient to Rome, but their first job was to act as a buffer if an enemy wanted to attack Rome. The idea was that they would have to go through various client states first and that would weaken them Anyway back to the census. Then there is a cock and bull story about how they had to go to their ancestral homes. If you knew anything about a census at all you would see how that was ridiculous. Not only that Nazareth was not in Judea. It was under another of the sons of Herod at that time and Rome had not taken it over. They were not technically citizens of Judea any longer so there would have been no reason to go to a Judean census. Second people do not travel to ancestral homes for censuses that were designed for taxation purposes. No one cares where you came from. They care about where a person lives, and more important earns money. There is no valid reason for Joseph to go to Bethlehem to get counted. At that point the myth has already failed beyond repair.