Nowhere does the Constitution say that the Constitution shall be suspended in time of war. Nowhere*. With the Constitution always in force, a President has no choice but to continue to act within the constraints it specifies. A declaration of war does not grant extra-Constitutional powers.

It's as simple as that.

(*--A nitpicker might point to Article 1, Section 9, which grants Congress the power to suspend habeas corpus in "times of rebellion or when public safety requires it". That's often given as the basis for martial law, which is quite a stretch. But that's the sole reference to any suspension of any part of the Constitution. That's it, and it's not enough to build an Imperial Presidency on. See USConstitution.net for an interesting list of things people think are in the Constitution but actually are not.)