I love "my players keep out threads". They're so secretive!
How about a permutation on the "players sit in on the trial"? You've got some good guests out there, put live bodies on the committee. Surprise the players with them, and set the whole thing up like a formal trial. Remember PC's trial of Kiri?
Have one of the candidates threaten them with harm if they don't get the job. Or harm if they do get the job. Or have one bribe them, which ought to send Anvil right round the bend.
The real thing to consider is why are the PC's important? What about the role of Chancellor is important? Is he just a big talking head, or is he Gandalf/Dumbledore/Merlin all rolled into one (that's a lot of beard!)? If you answer those questions, you'll be a long way towards figuring out what to do.