A treaty carries with it force of law, so if Congress approves a treaty with powers in the realm of international commerce, it is also essentially passing a law on international commerce. That's the "Consent" part, and Fast Track appears Constitutional.
On the other hand, the Constitution does say "Advice" too. Fast Track would seem to do away with the "Advice" part (these days done through amendments, etc.), thus making Fast Track seem unconstitutional.
Also, looking at history, the power to make treaties was originally given to the Senate, and the president only added to the process just prior to ratification. Over the years, this has evolved into what it is today - an executive monopoly on negotiation.
It seems Fast Track would push the balance a bit too far to the executive, but it has been going in that direction since before the Constitution was even ratified..
Q, And NOW would be a GOOD time to put a stop to it! NO FAST TRACK!!!. By the way, the "free" trade agreements made thus far are NOT "treaties", but "agreements". "Treaties" are the responsibility of the President and the Senate only. Let these make and be responsible for their "mistakes" and advances too. There are only 101 of them to be held responsible rather than 436. Peace and love, George.