That is, the Law of The Excluded Middle (being one of the most basic axioms but just one concrete example) exists because that is the way the mind of God works. But as a principle it did not exist "before" or even in parallel with His existence. I do not believe it would be true if He thought Truth existed in some other way. The axioms all seem "natural" to us, because being in His image we think as much like Him as we are able in our most lucid moments.
Ah, but that is the whole point. The "Enlightenment" empirical worldview holds that reality is defined empirically - if it can't be discerned by the senses it doesn't really exist. This is the Richard Dawkins or Christopher Hitchens position. Whether you call it "empiricist", "naturalist", "materialist" or "modernist", - this is the common foundation of each. If math, logic, and even morals are independent, objective aspects of reality which exist apart from (or "transcend") the material, "natural" universe then that worldview is blown apart. The box is now open and you're back to Plato.
Don’t get me wrong on the Plato allusion. I definitely agree with you that these non material objective aspects of reality are a reflection of the mind of God. They couldn’t exist any other way.