Nonsense. They could declare it anything they wanted. Obama would sign it regardless. Obama and Rhodes used the Senate as part of the scam. They turned the treaty process on its head requiring only a one-third vote for approval.
If the Senate had not gone along with the scam, Obama would have bypassed the Senate entirely. The fix was in.
We will never know. What is however clear, is McConnell surrendered his Constitutional power to consider this scam a Treaty. The GOP went along. It would be a leap even for this SCOTUS that they would have overturned a direct Constitutional authority.
"Article II, Section 2 of the Constitution gives the President the power “to make Treaties, provided two thirds of the Senators present concur.” And the Supremacy Clause provides that “treaties,” like statutes, count as “the supreme law of the land.”"