So I bothered to make an entire hypothetical caretaker cabinet. As technocratic and corporate as you'd expect it to be, although I feel there ought to be a few more academics and a few less CEOs in this. I'm unsure whether to swap Sandberg (who I'm not at all crazy about) and Whitman:
Acting President: Robert M. Gates (R->I)
Acting Vice President: Sheryl Sandberg (D->I)
Secretary of State: Jon M. Huntsman Jr. (R)/Richard N. Haass (R->I)
Secretary of Defense: James Mattis (I)*
Secretary of the Treasury: Laurence D. Fink (D)
Attorney-General: Rod Rosenstein (I)**
Secretary of Commerce: Kenneth Chenault (D)
Secretary of Education: Michelle Rhee (D)
Secretary of Energy: Ash Carter (D)
Secretary of the Interior: Peter Fitzgerald (R)
Secretary of Housing and Urban Development: Patricia Harris (I?)
Secretary of Health and Human Services: Olympia Snowe (R->I)
Secretary of Transportation: Elaine Chao (R)*
Secretary of Veterans Affairs: Sloan D. Gibson (I?)
Secretary of Agriculture: Rosalind G. Brewer (I?)
Secretary of Labor: Gordon H. Smith (R)
Secretary of Homeland Security: John F. Kelly (I)*
Ambassador to the United Nations: Meghan O'Sullivan (I?)
National Security Adviser: (Lt. Gen.) H.R. McMaster (I)*
Director of National Intelligence: John O. Brennan (I)
Director of the National Economic Council: Meg Whitman (R->I)
*remaining in office (if not already dismissed)
**conditional on performance as Deputy AG
This seems like a cabinet that would really only be supported by professional Moderates (TM), and probably no one else. Why would Liberals want McMaster, Rosenstein and Huntsman in this? And does anyone actually care about Sheryl Sandberg?
If the lids actually blows off on the Russian connections, I think the caretaker would have to be someone who is already in the line of succession but not controversial. Sonny Perdue is the only one I can think of--two term governor, probably the most harmless member of the cabinet.
I think David Shulkin is even less controversial and actually well-liked by both sides.