Mary and Joesph were not refugees. At all. Nor were they poor.
They did not make travel arrangements very well.
It's not like they could hire an Airbnb, or make reservations at the inn over the inter webs.
...they were on a government boondoggle to make it to Bethlehem for the Roman required the census before January.
According to Google AI, the decree went out on Monday December 18, 3 B.C.
Their travel arrangements were as good as Augustus’ census made them.