Not sure about the difference in time for return of the funds between Debit Cards and Credit Cards. For Credit Cards, the company we use 'says' 1-3 days, but in reality it is usually within 8 business hours. I assume it is the same for the Debit Cards, but I have never had the need to check.
We don't distinguish between Debit and Credit cards for deposits... not sure about the other companies out there. We do put the security deposit 'hold' on funds. Depending on the length of rental it is either 10,000 or 20,000 pesos. (Maximum, just shy of $700USD by today's math).
I agree with the logic on using for security only. Paying in cash is the way to go.
I'm certain that we would be more expensive than the competition in Santiago, but if you come to POP or Caberete (or down to La Romana/Bayahibe), let me know and we'll try to help you out.
But to answer your question... most of the agencies down here probably will accept a debit card without any trouble. Some of the bigger US/International ones have special caveats if you use one with them.
Good Luck!