Here's where I think Cheney would be needed: cloture fails, the Republicans raise a point of order (Nuclear Option) that only a majority of the 100 is required for cloture on votes to confirm nominations. The parliamentarian sustains the point of order & the Dems appeal that ruling to the full Senate. The Senate votes 50 - 50, The VP in his official capacity as President of the Senate casts the tie-breaking vote & only a majority is required for cloture thereafter (until the rule is changed again.)
Oh, I just had a thought. If the sillybuster DOES happen, the Republicans can't/shouldn't pull the trigger on the constitutional option today, because they will need Ensign's vote on the question of using the fillibutster on judicial nominees. If there are only 99 senators available to vote, Cheney wouldn't be needed to 'break a tie'. The Republicans may have to wait until tomorrow to put the rule to the chair, assuming Ensign will be there by then.