Michael was disgusted with himself how he hadn't seen through Roth's deceit including Roth's blessing!
You see this in the Havana scene where he and Fredo are having their drinks. He tells Fredo that he'll be assassinated before he reaches his hotel. Fredo says, "Who?" Michael says, "Roth," and he says it with disgust--IMO, not only for the idea that he'd be assassinated but also because he'd fallen for Roth's line of BS until the shooting.
I believe, the reason Michael sent Frankie to the Rosatos' meeting was to continue to appease and string Roth along including Michael had not twigged, it was Roth all along and also to find out the traitor in the family not “to put him [Frankie] to a final test†nor Michael needed to be “convinced it wasn't Frankieâ€
Hardly. The last thing Michael said to Roth in Miami was, "Frank Pentangeli is a dead man--you don't object?" But, instead of having Frankie killed, he instructs him to have the meeting with the Rosato brothers. Roth should have been alarmed by that because it signaled that Michael didn't suspect Frankie in the shooting after all--meaning that he must be suspecting someone else. Also, Roth should have been suspicious that Michael asked if Roth would object to his having Frankie killed--why would Don Michael ask Roth's permission to kill a subordinate who,supposedly, was responsible for the Tahoe shooting? It was as bad as Michael failing to question why Roth interested himself in the Rosatos' beef with Frankie over three territories in the Bronx. Roth was slippin'...
