I really like Tissot’s painting because it looks just like Old Jerusalem today.
The word for ‘inn’ in this verse is ‘kataluma’ which can be translated ‘guest room’. The standard housing design had a guest room. Basically, none of Joseph’s relatives would let them into their homes, perhaps because they were considered ‘unclean’ because of Mary’s ‘illegitimate pregnancy.
When I was in Israel, the speculation was that they were in a shepherd’s cave or barn. Another possibility is they were in someone’s home in the barn portion which would be under the main living portion of the house.
It’ll be fun to get all these details straightened out after Jesus’ return.
I like your speculation about rigidly-prudish relatives, although one can certainly imagine that a large census-driven influx made for space at a premium. But to refuse any woman in labor....