Jews dont travel en masse and fill up all the inns in December. They do, however, do so for their spring and fall holy days. Which would be mid-late September or Early October for the Fall ones, or Aprilish for the Spring ones. Additionally, shepherds dont hang out in fields in December. Unless they enjoy freezing their butts off and getting their sheep sick. So take your pick - whenever He was born, it was no where near December.