Those men were men of great integrity and principle. If they had thought that their signature on the Constitution could reasonably be taken as an acceptance by them of what they did not in fact believe, they would have refused to sign the document.
This idea you have, that the usage of a boilerplate phrase that mentions "Our Lord," necessarily implies full acceptance of all the connotations and assumptions that anyone might be able to deduce by dishonestly focusing on the literal meaning of the phrase, while ignoring the normative meaning of the phrase in its sociocultural context, is simply absurd.
The existence of the phrase "In the Year of the Lord" in the English language proves that the speakers of English have a society and culture that has been influenced by the Christian religion. That is all it proves.
Even if all the signers of the Constutition believed in God (and in the same God, for monotheism does not imply agreement about the nature or identity of the one God, nor about the identity or content of his revelation(s) to mankind), such a fact would have no relevance to the proper interpretation of the First Ammendment, nor would it have any bearing on the merits of whether or not it was Constitutional to have a law that requires government employees to lead school children in a recital of the 1954 version of the Pledge of Allegiance.
Let me see if I read this right.
Our society and culture was influenced by a religion and that the composition of the declaration of independence and the constitution were also influenced by this same religion. Did I read this right? Not trying to be nasty, just to understand your view.