Treaty obligations do not have "almost constitutional standing". Treaty obligations have exactly the same standing as any law passed by Congress, whether they're constitutionally ratified (which is extremely rare), or congressional-executive (the common kind), in which the president signs a treaty that is completely non-binding on the US, but promises that Congress will enact legislation to fulfill its terms[0].

In either case, there is zero domestic legal consequence to deciding to ignore the treaty provisions. Technically a decision by the executive branch to violate treaty provisions is a violation of federal law, but there are no consequences. Obviously, if the US decides to violate a treaty agreement there may well be international consequences, including other nations' decisions to back out of the treaty, to refuse to trust the US in other negotiations or treaties, etc. All of that, of course, is exactly the same regardless of whether or not there is a formal process of derogation.

[0] There's a third form of treaty, "sole executive". That form is applicable when the treaty obligations are entirely within the scope of presidential power, such as Status Of Forces Agreements which the president executes in his role as Commander-In-Chief. Those, of course, have no legal standing at all except as lawful orders from the president to his employees.