I doubt the customer service desk will even accept your $10. Customer service representatives at larger corporations like Walmart are instructed to appreciate any and all business with each of their customers. It would ruin the store's credibility if they made a mistake and then request that their customer's pay for it.
Plus, if you went in with a receipt and paid them back, the most harm would be done to the employee that made the mistake. They could be dismissed from their shift, told to go home early (missing out on valuable pay), or even terminated if it is one of many previous mistakes the employee has made.
Do not go back into the store. Ultimately, it would be wasting your time, customer service management's time, and the employee's time (and maybe wage, or even the job itself)
$10 to any single Walmart in any given day is absolutely marginal and would make no difference, whatsoever, to the department. I have a friend that works in financing at a larger Walmart near my town and he claims that every Walmart rakes in anywhere from $15,000-$30,000 dollars in profit alone every single day.