The answer is: nothing. Our dating system is an historical artifact of Western culture, and has no legal significance or implications for the meaning of the Constitution or the First Amendment. The American Colonies were established by Europeans; we naturally inherited the European practice of dating years from the birth of Christ. Nothing follows from this except the trivial observation that, in establishing our independence, we decided not to completely overthrow our cultural heritage.
In fact, the European dating system is infused with pagan holdovers that, if taken seriously, lead to exactly the opposite conclusions reached by accommodationists. We have a seven day week, after the model of ancient Israel, but we inherited Pagan names for these days; does the Constitution then establish Sun worship when it excepts Sunday from the ten days Presidents have to veto a bill before it becomes law? Does it establish worship of the Moon when it says that Congress will begin it's sessions on the first Monday of December? Does the use of European names for months mean that the Constitution establishes worship of Julius Caesar (July) or Augustus Caesar (August)? The issue was a serious one for some Christians; Quakers, for example, adopted numerical references for days and months precisely to avoid objectionable Pagan names. The rejection of the Quaker system suggests that the founders read very little into their dating practices. To base an argument on those practices is to stand on extraordinarily shaky ground.
To be sure, the Constitution could have avoided the words "Year of our Lord" in the date (as it does elsewhere when it refers to specific years), but it's hard to imagine why. "The Year of our Lord" was the standard way of dating important documents in the 1700s; its use was ritualistic, not religious. It is doubtful that anyone, Christian, deist, or otherwise, would have given the words a second thought, or ascribed to them any legal significance. And if the intent of the Constitution was to signal a favored status for Christianity, it could have done so in a thousand less ambiguous ways than including the words "in the Year of our Lord." That some accommodationists appeal to these words is silent testimony to how little evidence there is for the idea that the Constitution embodies Christian morality or thought.