I'm guessing that Michael knew immediately after the shooting that he had to leave the Tahoe estate at once. Reason: he would have figured out that he, not Kay, was the target; but the man behind the shooting was ruthless enough to not care if Kay or even their kids were "collateral damage." He also knew, right away, that there had to be a traitor close to him who helped set it up. As long as the traitor and/or perpetrator were at large, and as long as he remained at the estate, his family was at risk.

So, I'm guessing, he zeroed in on Roth during the train ride. He called on Pentangeli, I'm guessing, to put him to a final test--asking him to stick his head in the lion's mouth by meeting with the Rosatos. Once Frankie agreed, Michael was convinced it wasn't Frankie--it was Roth.