The House is not involved in this because this is a senate only function in the constitution, as far as I know.
So, if this "deal" is not legislation but just an informal agreement among friends then the Senate could change its mind at any time and rescind it.
Regardless, they cannot legislate away a power. Only an amendment can do that.
This stinks of McConnell. It's the same deal he made for the debt increase that undermined Boehner in the House - allow it with an unreachable override clause.
Since the Senate is the body that represents the states, the states will have to sue on the grounds that it violates the Article IV guarantee of a republican form of government. The Senate must uphold it's treaty power. It cannot twist it into a treaty super-majority veto override power.
-PJ