"[...] But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States."

Didn't look at the 12th because I was focused on the 22nd.

But if we look at a careful reading of the 12th, 22nd, and Article 2 (qualifications for presidency):

"No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States."

It will all come down to the question: does the 22nd Amendment make a 2 term president ineligible for the office, or make that president only ineligible to be elected to the office?

It could be argued either way, but the way the 22nd is phrased leads me towards the latter interpretation.