I don't think that would work so well, since you could never prove how many intermediary cash-only transactions the bill has gone through before it was seen by the bank again.
Agreed, for a single transaction it would not meet the "beyond a reasonable doubt" threshold. However it might be possible to build a stronger case around a pattern of transactions, e.g. bills withdrawn from the ATM by Alice routinely appear in bank deposits by Bob. At some point this would harder to explain as random chance.
