No, the bill would and does go to the floor as a bill.
Only treaties require more than a simple majority.
If something is not a treaty, it is not a treaty.
I really don’t see any logic to your words. It’s apparent you don’t understand what you are saying.
The first being a supermajority vote by the Senate.
The second is a Congressional-executive agreement. This is also known as Trade Promotion Authority or fast track. Once both houses of congress have authorized TPA, or fast track, a trade agreement (treaty) may become law with a simple majority of both houses of congress and executive branch ratification.
It really doesnt require much logic to understand this. Obviously the first route is much more difficult to obtain and likely impossible with the current Senate.