Spread the new religion beyond Rome earlier
Christian scriptures do not include a date or even a season for Jesus's birth. December 25 was a ret-con, established by the western church in the 4th century and chosen to coincide with the Roman solstice festival. The early church overlaid Christmas on a Roman festival because Rome was a huge concern in their lives.
The 4th century is pretty late, though. So don't wait until then. In the first decades of the existence of this new religion, spread its message among other nations, like Greece. Ancient Greece had a lot of festivals; choose one that has the thematic or timing elements you prefer, one that's not in winter. Establish the church early, and by the time church councils are meeting in later centuries, the date will already be fixed.
The church is still going to need to deal with Rome, but that's ok -- there are other events, undated in Christian scriptures, that could be overlaid on Roman festivals. The church could designate a date in late December as the day commemorating a significant miracle, perhaps the one about bringing a man back from the dead or the one about feeding a crowd with a few loaves and fishes. Rebirth in the depth of winter and feeding people in times of agricultural scarcity are both themes that could catch on. So you still get a holiday in December, but one that's clearly not as important as either the birth or death/resurrection of Jesus. The later acretions -- gift-giving, bigger festivities, secular hoopla -- would move to the new date for Christmas.
Santa might need to trade his big fur coat in for a beach towel and sunglasses, but people are remarkably resilient about these things.