I think Barzini only tricked the Corleones before the meeting of the families. After the meeting, it was clear to everyone that Barzini was the new top dog and I don't think he would still pretend otherwise. Tessio even mentioned that Barzini's men were taking over his turf which meant that Barzini wasn't secretly moving against them anymore.

By then, it were the Corleones themselves who were pretending to be weak. Michael denied Tessio and Clemenza to recruit more men so that Barzini and Tattaglia would think Michael would no longer be a threat, which was exactly what Michael wanted so he could carry out his plan by eliminating them.