You will not find any Car Rental companies who will accept a "Pre-Pay" Debit Card for the Deposit Requirement on a car rental. Some won't even accept a regular Debit Card tied to a regular bank account.
They either require an actual Credit Card(sometimes a regular Debit Card) with enough credit available to "hold". If you don't have that some will allow a Cash Deposit, but for that they often require a lot more additional information and verification. In some cases you can only do this if you notify them in advance.
If you want to rent a car then get a REAL credit card. If you can't even qualify for a credit card then why should a company risk loaning you an item worth several thousand dollars? If it is because you CHOOSE to not get a credit card, then that is YOUR choice. You are going to find several problems with these type of "Pre-Pay" cards.
By the way having the refund available only after 24 hours is quick, sometimes it could take up to 2-3 days. You pre-paid for a reservation and did not read the rental deposit requirements, as had you read those you would have seen that they do not accept those cards. Read the last sentence of the terms directly from the Budget site(I bolded it to make it quite clear for you).
http://www.budget.com/budgetWeb/html/en/customer/commonquestions
"Deposit" credential requirements
The preferred second required credential is a major credit card such as American Express, Budget Charge Card, Budget International, Discover, JCB/Japan Card, MasterCard, Optima, PHH and Visa. Some locations will accept a debit card instead of a credit card, but then we might either check your credit report or require you to present a third form of ID as indicated in the “Debit Card User Instructions” below. Prepaid charge/debit or gift cards are not an acceptable form of identification at rental pick up but may be used for payment at car return
