But the problem with allowing religion to then influence the government except in the barest of ways is because then the government is favoring one religion over another.
Thus, it is a two-way street.
If an amendment is passed saying god is the supreme authority above the law, and god's law holds precedent, that means that religion is influencing the government in a manner that favors one religion (well three technically, Judaism, Christianity, and Islam) over all others, and leading to that one religion being used in legal matters which can interferre with other religions.
Then there's the mess of interpretation. Some will claim that it's illegal to eat shellfish due to this law, and you should be stoned. Afterall, since god's law overrides murder is now possible. Other judges could rule that it's irrelevant due to the new testament or koran. Yet still, other judges could say things like gay marriage are permissable since the new testament voices no objections against it just as there are no objections to shellfish.
So overall, this is a law that does not benefit even the most pious, as there are more loopholes in religious law than even common law. Thus why there are so many different sects...
Then there's the issue of who's god is god? By declaring in law that god is Allah, Jehovah, the Trinity, Cthulu or whatnot you are then by definition forming a state religion. By not declaring by law who is god, then you're in the interpretation issue which means that the law will vary greatly from judge to judge. Far more so than the common law system. Legal systems need to be concise and consistent to operate properly.
So, the logical execution of this law is unconstitutional, despite the fact that the wording may permit it to be constitutional for some.