As far as I am aware (and I'm happy to be corrected) there was never a census in the Roman world at that time, before and after perhaps. The Romans kept very good records and there is none.
Since the primary documentation of the Christ events were first written in the book called the Gospel of Mark, Luke and Matthew had been included as corroborative evidence although perversely as you point out they are much at odds with each other!
Luke and Matthew seem to be incorporating their own independent threads of variations on the traditional virgin birth based on pre-christian stories of the "saviour figure". These include the Mithraic legends of Persian origin, based on ancient astrological lore of the birth of the Son of God in the East who were visited by the three wise men (magi or persian astrologers). May I add that the astrological basis for this was that the three stars on the belt of Orion (the three wise men) point down towards Sirius the brightest star, hovering just over the horizon in the east at the winter solstice. I checked the night sky at Christmas and happily confirm this is still the case!
The "slaughter of the innocents" has no historical record or basis and not even the opponents of Herod subscribed to it. Josephus, a writer of Herod's life and would most certainly have spelled it out, never mentions it. There are however precedents in a number of the earlier saviour myths. Luke's account for example recalls (for a Roman audience) the legend of Romulus and Remus where their births were subject to a Herodian type of decree to slaughter all of the new-borns to deny the heroes a life.
Whatever the biased Watchtower says in an attempt to synchronise these dubious events; do not believe the Bible as literal truth. It is myth layered upon myth.