Most Muslims believe that they are required to follow the government they are under AND to live under Muslim law in regards to the Mosque. They believe that the Koran teaches, and actually orders, a separation of Church and State.Where does the Qu'ran specifically state that Muslims must live under an Islamic government? Most Muslims don't believe that's true: it sounds as if you're claiming to know more about Islam than the majority of Muslims.
But, as I said before, this is beside the point.
Whether the constitution is "necessarily at odds" with the teachings of the bible is immaterial. Some interpretations of the bible will inevitably be at odds with the constitution . . . or any other system of secular laws. When these conflicts arise, to which is your primary loyalty? Which takes precedence? If it's not the constitution then you can't honestly serve as an elected official in our government.