When was Jesus born?
The date of my birth doesn't correspond to the calendar that now measures your time; there was another way of measuring it. Even the way to measure days was different. In an approximation to the current calendar, it corresponds to the month of July, around the 24th, of the year 6 before the Christian era.
On the traditional celebration of Christmas, you can continue the drunkenness, but no longer in My Name. I have announced the date of My Birth; it wasn't the 25th of December.
Can the feast of Sol Invictus and Santa Claus continue being compatible with December 25th?
Yes, you may continue celebrating what you want, but with the knowledge of the true intention, not with the instructions of marketing.
What is the real intention to be given to that day?
It's what marketing does: induce people to consume, hence the invention of Christmas gifts, of Santa Claus, the Christmas dinner, the massive sale of turkeys, etc. It is the success of commerce. The people should know that they are celebrating trade; that is the true intention.
Yes, they can continue celebrating what they want but calling it by its name, with its intention.