The 22nd only prevents him for getting *elected* more than twice - could he not run as a VP with another presidential candidate and then have the puppet president step down?
Discussion
But that's the thing -- he's not ineligible to be president, simply ineligible to be elected again.
https://www.whitehousetransitionproject.org/wp-content/uploads/2016/04/Terms-Tenure_101909-1.pdf
>The debate continues as to whether the 22nd Amendment is an absolute limit on service as President. For instance, could a former President elected to two terms later be elected Vice President, and then succeed to the presidency on the departure of the incumbent chief executive?