It doesn't trump the Constitution, the Supreme Court has previsouly determined that Treaties cannot act in contravention of the Consitution, rather, the Constitution gives the weight of national law to treaties, which includes the Vienna Convention and the ICJ statute.
But giving a treaty the scope of national law, in this instance, does not simply specify law in an area that is already under federal jusrisdiction (like entry to ports or foreign trade). It extends it into interfering with state sovereignty with an authority that I would think even the Congress (if not the SCOTUS) would blush to assert.