The criteria used by the IRS suggest that for federal tax purposes, the cashier would be properly classified as a contractor.
Behavioral Control: A worker is an employee when the business has the right to direct and control the work performed by the worker, even if that right is not exercised.
There is probably some training involved here, but it seems negligible. Has the company actually retained the right to direct and control the cashier? Is he obligated to use their register, as they tell him to use it? Is his work evaluated for compliance with those instructions? I'd guess that in most cases, the company doesn't actually care about any of this. As long as the company got all the money it was owed, would it really care if the cashier just stuck the money in his pockets until the end of the day?
I'd also argue that it isn't really the employer controlling the when and where of the contractor's work, but rather the circumstances. Classifying him as an employee because he has to be at the event doesn't really make any more sense than saying your plumber is an employee because he has to come to your house to do the job.
Financial Control: Does the business have a right to direct or control the financial and business aspects of the worker's job?
The investment in the cash register seems relatively nominal. The company probably does not reimburse the cashier for expenses incurred in getting the job done. The cashier is presumably free to offer his services to others. The cashier is presumably being paid a one-time, flat fee.
As you noted, though, the fact that there isn't much profit-loss opportunity is one factor pointing in the other direction.
Relationship: The type of relationship depends upon how the worker and business perceive their interaction with one another.
The fact that this is a single, hours-long job is probably the strongest evidence that the cashier is a contractor. Further, the cashier's job is not a key part of the business, as it is only a minor portion of an event that the company has never performed before and has no apparent intention of repeating. I assume that the company is not providing health benefits, sick time, etc., and that any contract with the cashier includes no language suggestive of an employer-employee relationship.
Conclusion: The employee-contractor distinction is pretty fact-intensive, but based on what you've provided, there seems to be a much stronger argument that the cashier would be a contractor.