What authority besides the Supreme Court should we use to determine constitutionality?
How about we the people and the plain language of the constitution?
Try looking at the words of the Constitution. The word "treaty" is there. The words "trade agreements" are not there.
Every other nation who has signed off on this "agreement" refers to the "agreement" as a "treaty."
So the other countries are entering into a "treaty" and the United States is entering into a "trade agreement."
Every citizen, and every officer of government who is required by Article Six to take the oath, has to determine constitutionality. If they won't, they are either a poor citizen, or a sworn officer who cannot possibly keep the oath and provide appropriate checks and balances within our form of government.
"Constitutions are not designed for metaphysical or logical subtleties, for niceties of expression, for critical propriety, for elaborate shades of meaning, or for the exercise of philosophical acuteness or judicial research. They are instruments of a practical nature, founded on the common business of human life, adapted to common wants, designed for common use, and fitted for common understandings."-- Joseph Story, Commentaries on the Constitution of the United States
"We the people are the rightful masters of both Congress and the courts, not to overthrow the Constitution but to overthrow the men who pervert the Constitution."
-- Abraham Lincoln