Our church outright teaches it was most likely Sept 11th, either 6BC or 8BC and the latest of 4BC.

The census was not a single-day event, and the "date" was the deadline, not the start of it. Many governors would have started the census years before the deadline. Also, Luke ch2 is the only mention of a census, and could have been a literary device rather than a hard fact since about 10 years later there was a revolt in reaction to the census, for which the Holy Family wouldn't have even been present.

Herod the Great died 4BC. The "wise men" visited when Christ was between 1 and 2. This is when Joseph fled to Egypt, after the wise men told Herod about Christ. The "star" they followed was taught to them in Persia by Daniel, during a certain alignment of the planets and constellations that occurs about every 800 years, which happened around 6BC.