I am sure that someone could eventually pin it down. But, the point of the matter is that no one can definitively tell you when this baby, born to an out of wedlock mother and his carpenter father, was exactly.
Well, unless there is some undiscovered and new information that we do not have right now, I don't know how anyone could pin it down any more than it is right now. We do have a lot of clues as to the time of the year, but I don't think that's ever going to get us close to a particular day of the year.
And there are reasons why God left some details out, while including other details, in the Bible.
Think about it in our context. A baby is born behind a garage in a shed, because its homeless parents cannot afford rent.
I don't think that those parents were ever really "homeless" though. I mean, being unable to find location accommodations on a trip that one is making is hardly being homeless... :-)
They had a home before they left on the trip and they had a home after they made the trip. The crowded "inn conditions" at that particular time, hardly makes one homeless. They were unable to come up with the kind of accommodations that they had sought at the time, as that was all they could find, but to say that they were homeless is to depict them in a light that they really weren't in, in their situation.
But, aside from the issue of "homeless," it wouldn't bear on the date of the Messiah of Israel's birth.
It seems fairly clear to me that the general time period of the 25th of Kislev being the time of conception, would fit very well what we know of the situation.
It was April 6th.
Mary and Joseph were going to Bethlehem to pay their taxes.
I believe it is left out as a Divine lesson upon His return.
Just as we know only the season, and a small range of years for His birth; so, too, do we only have clues as to the season, but not the year or time of His return, either.
Had either or both been foretold in full exactness, then, like Obama setting an Afghanistan withdrawal date, sinners could just 'play out the clock', then "repent" just before His arrival.
Depending upon a deathbed conversion is a dangerous tactic to attempt by the unrepentant. Just as we know not the hour of His coming, they know not the hour of their departure.