Christianity has no quarrel with the Constitution whatsoever. The vast majority of the Founders and those who signed the Constitution were in fact Christians, and yet they were determined that Christianity not be made the official religion of the land and no pope or monarch or any other central government would have any power to dictate what any person must believe in anything. That was their definition of liberty.
Sharia Law would definitely include giving government the power to dictate the religious rules the people are required to live by and to inhibit or squelch all other beliefs.
Christianity has Freedom of Religion? Of Speech? Of Action? Of Privacy and Sexual relations? It has voting? It has Public Service? It has Fair and Speedy trials? Those are Enlightenment Ideals. Christianity doesn't have a one of them. Christianity has God and the submission to God, and Hell for those who reject Him. Just like Democracy and Capitalism, Christianity is incompatible and unrelated.
When the Catholic Church was running Europe, that was Christianity as a Government. Sharia would be very close to the same.