You are looking for 24 person-nights, so it comes out at $62.50 per person per night. I think that's pretty low if you want food included, given that most places that include food also include drinks. I don't think that even a really basic hotel like Hamaca in Boca Chica would offer you that rate. We have discussed AI rates to death on another thread and I think they are currently running at around US$ 90 per person per night, lowest.
You would certainly get a nice room for maybe $90 per night, including tax, in one of the better Colonial Zone hotels - like Luca or Casa Sanchez or Palacio, and that rate would include breakfast. That would leave you $35 for lunch and dinner, for four people, RD$ 2,080 pesos for 8 meals. Just do-able if you eat in cheap places, but I think you'd probably spend more.
I would advise you to email the hotels and discuss rates. Hotels have to pay commission rates of about 20% to the main web pages, and are not allowed to offer lower rates on their own sites. But they can offer lower rates if you call, email or walk-in. And they will usually give you at least half of the 20% commission they would otherwise be paying as a discount.
Final thing to say is that hotels are, surprisingly, very busy at the moment. My hotel has been 100% full all week and looks like it's staying that way. Rates are high as hotels are trying to make up for the huge amount of money they lost last year. So don't expect rock-bottom rates.