I don't believe a Wizard number on Avis will work on Budget side. You will not get the advantages of a stored card or pre-reserved assignment unless you have a Budget Fastbreak ID as well. For a shared location, probably not an issue, but there is no reason to not have a Budget ID as well. On a side note, I find the Budget Bucks program probably better than any Avis points program out there and 'stacks' on top of practically all types of reservation coupons and promos.
Rasheed