Hmm...
Since each person would get to replace a judge one time, the other person could not refuse this or insist upon keeping the original/replaced judge.
So...
Warren and Kay fight over Shukaku. Judge is Asadi.
Warren replaces Asadi with Koji....Asadi can never be used again for this match.
Kay replaces Koji with Bocchiere...Koji can never be used again for this match and we are stuck with Bocch...
However...maybe:
Warren proposes Koji to replace Asadi but Kay says no. You have to agree upon a new judge. If you cannot agree upon a new judge, the council steps in to appoint one?
Or the council just bans both Kay and Warren from bijuu activity for 3 months for being stiff necked jerks?
This could force a person to compromise and stop deadlocking these events?
Either way, we are fighting...we have one judge. a difficulty arises, we get a second judge. more difficulties arise. We get a third judge. I think that is plenty enough leeway toward resolving the issue. If we can't manage to conclude a match after going through 3 judges then clearly either Warren or Kay or both of them are fuck ups and should be banned for 3 months so they can get their shit together and stop creating this mess.