Can a two-term US president secure a third term by running as Vice President?

From a constitutional perspective, I find a different question much more interesting:

What method does the Constitution provide to handle a situation when a sitting President, with the support of the DOJ and FBI, refuses to leave after his term has expired?

Or a sitting President who is impeached by the House and Convicted by the Senate, but who, with the full backing of the DOJ and FBI, refuses to step aside?

Our current SCOTUS majority would probably respond:

"Well I dunno, even if it's wrong, let's let it go indefinitely while it winds its way through the courts ... especially if it enables our guy ..."

tl;dr From the 12th Amendment:

> No person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.

https://www.archives.gov/founding-docs/amendments-11-27#toc-...