What's the difference? He can't be elected President, just like someone under 35 or someone born outside the United States. Doesn't that make him constitutionally ineligible?
No, it doesn’t. It makes him ineligible to be elected, not to hold the office.
The constitutional elegibility requirements are the same as they ever were.
It would be an interesting test case, but just following the letter of the Constitution I would say that he could run for the VP spot just the same as he could run for Rep, become speaker and be 3rd in line.