"However, the President's authority as Commander-in-Chief is not unlimited and US law dictates that the attack must be lawful and that military officers are required to refuse to execute unlawful orders, such as those that violate the Laws of Armed Conflict."

Those laws are very very vague and like you said, it's possible someone in the chain of command could argue it's against the law since a nuke would likely cause collateral damage one way or another. I think it's technically incorrect to definitively say that the US president has complete and total control over nuking a target.