There are several problems with this challenge. 
 
First, the Catholic Church celebrates Jesus’ birth on December 25, but this is a matter of custom rather than doctrine. It is not Church teaching that this is when Jesus was born (note that the matter isn’t even mentioned in the Catechism of the Catholic Church). 
 
Second, although most Christians today celebrate Christ’s birth on December 25, this was not the only date proposed. Around A.D. 194, Clement of Alexandria stated Christ was born November 18. Other early proposals included January 10, April 19 or 20, and May 20 (Jack Finegan, Handbook of Biblical Chronology, 2nd ed., §488, §553). By far the most common proposals, however, were January 6 (ibid., §§554–61) and December 25 (ibid., §§562–68). 
 
While the last was eventually adopted by the Catholic Church for use in its liturgy, the fact that the Church did not declare alternate proposals heretical shows the matter was not considered essential to the Faith. 
 
Third, the proposals that put Jesus’ birth in the colder part of the year (November 18, December 25, January 6, and January 10) are not ruled out by the fact that there were shepherds keeping watch over their flocks at night. 
 
Ancient Jews did not have large indoor spaces for housing sheep. Flocks were kept outdoors during winter in Judaea, as they are elsewhere in the world today, including in places where snow is common (search for “winter sheep care” on the Internet). Sheep are adapted to life outdoors. That’s why they have wool, which keeps body heat in and moisture out. 
 
Sheep are kept outdoors in winter in Israel today: “William Hendricksen quotes a letter dated Jan. 16, 1967, received from the New Testament scholar Harry Mulder, then teaching in Beirut, in which the latter tells of being in Shepherd Field at Bethlehem on the just passed Christmas Eve, and says: ‘Right near us a few flocks of sheep were nestled. Even the lambs were not lacking. . . . It is therefore definitely not impossible that the Lord Jesus was born in December’” (ibid., §569).

One of the difficulties in understanding why Jesus suffered for sinners is that many of us, especially in the modern world, tend to think of sin only in legal terms as the breaking of some abstract rules. But sin causes real harm, and has real effects, and these must be healed. Something actually has to happen. 
 
For example, let us say that you see me near the edge of a cliff and I warn you to take three steps to the right. But let us also say that, out of rebellion, I take three steps to the left, and slide down a great cliffside into an abyss. I lie there, injured and utterly incapable of ever rescuing myself. Let us then say, in my humiliation and pain, I cry out and ask you to forgive me. In your mercy, you say yes. And that is forgiveness. But in order to be healed and restored, you will now have to expend great effort to come down the cliffside, care for my wounds, and carry me out of the abyss. 
 
As I hope you can see by this analogy, my sin was not simply the breaking of rule. Deep and devastating effects happened in my life, and I was incapable of restoring myself. And this was our state; we were dead in our sins (Col. 2:13). We were incapable of ever restoring or healing ourselves. 
 
Jesus therefore, not only brought us God's forgiveness, but also extended the effort and agony to come down, heal our wounds and lift us up. This was a great, and painful, effort. Our sinful disobedience had brought us suffering and death. Jesus took up that suffering and that death in order to restore us, even elevate us to a higher place. 
 
The horrible suffering of Jesus shows us very clearly how awful sin really is, how it disfigures, wounds, and even brings us death. These are realities, and Jesus takes them up in order to heal them and carry them away for us. We tend to make light of sin today. It is no light matter, and to remember that, we do well to look to any crucifix and see what love cost him.