The morals of most secular people in the West are Christian in origin regardless of how much they may want to deny it. Secular humanism is just a version of Christianity without God, for the most part. Atheists and agnostics in the West are quite different from the same groups in places like Turkey, for example.
So, you can't really divorce the Enlightenment from Christianity. The Enlightenment itself wouldn't exist, were it not for Christianity. There's a good reason it didn't derive from the Islamic World, for example.
It doesn't "betray the sanctity of Christianity" to connect the Constitution with Christianity. If anything, denying its influence downplays how important Christianity has been for the development of the US. And the way that this society has declined in morality is precisely because of how much mainstream society wants to discard that influence in favor of materialism, collectivism, race Marxism, gender theory, feminism, and the LGBT nonsense. It's fallen precisely because of how we have let our government, media, and academia dilute and discard faith.