I don't think that there is anything in the Constitution that would prevent him from continuing to be Acting President until Jan. 20, 2021 (assuming he doesn't die of VD or a cocaine overdose in the meantime).
I think when Kissinger and Albright held ‘high’ Cabinet positions they were disqualified from becoming Pres because of birth/citizenship issues -
Be interesting to see how that would be handled today??
Like an earlier poster said you can’t ‘backdoor’ it if you are Constitutionally ineligible on one hand.