I disagree. The 22nd Amendment doesn't prevent anyone from becoming President more than twice; it bars them from being elected more than twice:
Section 1. No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. . . .
Since he wouldn't be elected President a third time, he wouldn't be in violation of the Constitution.
Have you got it now?