How did Don Corleone know that whoever set up this meeting was the traitor? And if he knew how would Barzini not know that the Don would know and warn Michael?

I don't think he knows with certainty, I just think he knows that Barzini works through discontented people close to his targets while appearing innocent himself. I think Barzini is supposed to be overconfident because he assumes Vito is past his prime and Michael is not an experienced leader.

...because Vito has decades of experience dealing in his world of deception and death.

It was Barzini who set up the meeting with Tattaglia and the other families.

"I didn't know until this very day that it was Barzini all along"

It was the same scenario with Barzini setting the meeting with Tessio.

In that world you're suspicious of everyone, including those that are close to you. More so probably because they can hurt you the most because they are close. If any of your top guys are meeting with the enemy you would be blind and stupid not to be aware of what could happen. So his antenna is already up and looking for trouble. With Barzini's past it's pretty easy to put 2 and 2 together if you're paying attention.

As Yogi Bera once said "You can observe a lot, by just watching"

Barzini helped Vito arrange the meeting. Barzini sat at the head of the table. Barzini announced the "agreement" on drugs even before Vito agreed to it. I wonder why Vito didn't know before the meeting that it was Barzini all along. The novel tells us Barzini was younger than the other Dons, regarded them as "Moustache Petes," was impatient with being the Number Two Don...and didn't Vito know all along, before the meeting, that Tattaglia was a pimp; alone he could never have outfought Santino?

As for Barzini: he made the mistake common in the trilogy of underestimating his enemy. Vito underestimated Solozzo; Sol and McCluskey underestimated Michael; Michael underestimated Roth; Roth underestimated Michael; Zasa underestimated Vincent; Michael underestimated Altobello...

It's not that Barzini set up the meeting. After all, Vito initiated the meeting in the first place. But Vito knew that someone other than Tattaglia was backing Sollozzo.

I think that Vito went into the meeting with two goals:

1. To get the other Families to agree not to deal drugs. Vito realizes this is futile after Don Zaluchi's speech.

and

2. To find out who was behind Sollozzo/Tattaglia. When Barzini took over the meeting and aired grievances, Vito knew that he was the secret adversary.

Some apples and oranges. The thread is about Barzini corrupting Tessio. Vito predicted an aspect of it due to his long time experience dealing with such situations.

The thread has morphed into discussion of the Dons' meeting. Woltz, you may be right. However, Vito's primary objective was to create an environment in which Michael could return to the US in relative safety

Yes, Oli, that was his primary objective. But, I think he also hoped to find out who was the prime mover. Still begs the question: why didn't he know it was Barzini before the meeting?

He could have meant that he wasn't sure it was Barzini.

Perhaps the important part of the phrase is "all along," that they knew Barzini had been supportive in the war like Cuneo and Stracchi, but not that he had been instigating Sollozzo and Tataglia from the start?

How he would gain that from the meeting I can only speculate that it might have come from Barzini's conduct at the meeting.

I reckon, the giveaways even without Vito's warning
1. Funeral is not the time and place to arrange meetings?
2. Normal procedure, Barzini's consigliere would approach Michael's consigliere
3. How and why rival Don and Capo were even talking, in the first place