You know the old joke - 4 Jews in a room with 5 opinions?
Most messianics take Sukkot as the birth time of Messiah. A sizable minority believe it was during Chag ha Matzot. But the truth is the Roman records of that census/ tax collection were stored in Pompeii and burned up when that volcano blew. So we cannot know for sure.
IOW, if it were important, G-d would have had the date preserved; but He did not. So I do not think it matters that much when (or even IF) we celebrate His birth. (which was not a classic Jewish tradition anyway)