You are right to have mentioned the fact that the date was fixed by emperor Constantine. In fact the circumstances described in the bible show that the day of his birth could not have been in winter.
For example shepherds tending to their sheep in the field or the census that was fixed during that time.
But the most important point is that we were never instructed to celebrate Christs birth. Christmas is an entirely pagan celebration that has nothing to do with Christ. It is definitely NOT worthy of celebration.