While it's true that Mormons are not one-dimensional and completely predictable, the odds of a successful relationship, given the OP's description of his girlfriend, are slim.
I married a person, not a religion. Response to your edit: Do NOT bring up or mention anything that could be considered "anti Mormon. ofo
If your doctor husband says "impossible", ignore him. Expect her to either write you off during her mission or pressure you to show interest in the church.