Was Christmas a replacement of the feast of Hanukkah or of the 'Invincible Sun'? Was Christmas a replacement of the Jewish festival of Hanukkah that was celebrated by the early Christians or was it to replace the Pagan festival of the 'Invincible Sun' instituted in 274 CE.
Why should Christmas be meant to replace anything? 

The liturgical year is riddled with merriment, every other day is a feast or a solemnity or at least an observance of some pious memorial. 

Other than to denounce Christ or the Churches that keep Christmas, there is no reason to conflate some pagan festivity no one cares about with Christmas.  If there was a conspiracy back in the fourth Century, then St. Augustine would have said something about it.  What he did say was that when the barbarians came to sack Rome, they left the churches alone, out of respect for traditions that were foreign to them, but which they could see were clearly holy.  They didn't see in them their own idolatry, they saw the city of God made manifest. 

Hanukkah on the other hand, has never been as big a thing as Christmas.  Christmas has been an important marker in the eyes of the Catholic Church purely out of necessity.  Consider the lectionary and how the Bible itself was formed.  The books we use at Mass were the same books and letters mentioned by St. Justin Martyr when he wrote about what it is that Christians do during their get togethers. Naturally, a book can't go on ad infinitum so it makes sense that at some point you're going to finish the book and start over again.  So, it makes sense that you'd want to put the events leading up to Jesus' birth in the first part of the book and go on from there.

So, we know Jesus died around Passover, we posit that the two most important truths in history took place on the same day of the year (The Incarnation and The Resurrection) and we do a +=9 months to get a good approximation of where Christmas should be. 

Pagans and Jews aside, the question to be asked is, is there really a better day for Christmas?