I like the idea of a FORMER Senator and Pence might be the guy. We need to remember, the VP has about as much power as a junior, a very junior Senator. One vote only in case of a tie. Other than that he/she is a good go between for the President and Congress and he/she needs to look good at funerals.
I may not agree with Pence much or any, but I doubt he would. like LBJ conspire to kill the top guy in order to get the keys to 1600 Pennsylvania Ave.
I am not certain the VP has no power in the Senate, has one ever asserted their presidency of the Senate or just by custom he only reports for tie votes.