Oli, the key word in your insightful post is "frustration." I think Michael was reveling in his Big Day: His Papal Knighthood finally "legitimized" him (as in Abbandando telling the news media, "The Pope has this day blessed Michael Corleone"). And then along comes Zasa, an urgent reminder of Michael's criminal side, asking permission to whack Vincent. And then along comes Vincent, telling Michael he should worry about Zasa because Michael is preventing him from rising in the Commission. So, I think Michael's initial reaction was to try to dismiss both Zasa and Vincent and get back to his "legitimacy," as in, "What does this have to do with me?" And, "So kill him." At that point, I think he would have been happy if they both went away and fought each other.

But, as you say, Vincent rang an alarm bell in Michael's mind. And, since Vincent showed his determination to act by biting Zasa, ear, the Old Manipulator figured that he might as well carry Vincent for awhile--use Vincent as a cat's paw for Zasa's evil intentions.