I could go on for pages about making a decision on where to live if staying at home is proving too expensive. I'll spare you the pages to wade through and take this one step at a time.
Family of four moving to the DR where husband needs to work and 2 kids need to go to school. Going to cost you in the range of $6000 USD plus misc expenses to get residency. Can't go to school or work here without residency.
You are probably not going to be satisfied putting your kids in public school here where they could conceivably know more than their teacher already. Lots of good private schools but I can't comment specifically about costs because I don't have any kids and schools are not a concern of mine.
Most people do not generally move from a place with a social safety net to a place that doesn't have one if they can avoid it unless they are financially secure (as in can afford medical evacuation home for life saving surgery secure). You will need medical insurance here for 4 people and that's going to cost money. Like the public schools, I think seeing your family in a public medical clinic will probably make you sick too.
I get that you expect to live a "minimalist" existence but what does that mean? Do you expect to have electricity, running water, air conditioning, food that your kids can identify? A can of Campbell's soup is more expensive here than it is where you are. I haven't seen any Dominican brand soup sold in cans yet.
Vehicles are insanely expensive here. A couple of years ago I looked at a 2002 Suzuki Sidekick. The seller was asking $5000 to start. I couldn't stop laughing long enough to even consider a counter offer.
Can anyone in your family speak Spanish? You can't effectively work here without being able to and your kids can't do well in public school without language skills. Everyone can learn, but life here is hard if you can't communicate. Most but not all native English speakers are retired or at least semi retired foreigners who choose to live in communities where they share some sort of connection with their neighbors. Not everyone mind you but the majority. There are not many newly arrived families making a go of it on their own out in the countryside without lots of money and a good command of the language. Money can pay for helpers to do the communicating for you but if on a tight budget, the amount of help you can afford to pay for will be limited. Simple English conversation with Dominicans can be found in the tourist areas, but as soon as you stray from hello, where is and how much is, things get progressively more difficult very quickly.
How much money do you need to live here? Good question. I can tell you what it costs me to live comfortably here and although that amount is less than at home, I'm not saving a bundle. I want familiar foods, a nice house with AC, a pool, a couple of dogs that eat $55 worth of dog food every 3 weeks. Minimalist to me would mean having to give up the dogs or the wife. Lots of debate regarding how much it really costs to live here. The answer depends on the lifestyle you need to maintain. Your idea of a good life might not exactly mesh with your wife's ideas and there is no hope that your kids will ever agree with either of you. For a family of 4 at $3000 per month would be really minimalist to me and you already know from above what I consider to be my voluntary bare bones requirements. This amount would not include incidentals, entertainment, and other non-essential expenses.
I'll stop here and conclude by saying, that your original post is devoid of any sort of factual reference points anyone can use to tailor advice to your specific circumstances. The fact that you even asked such a vague set of questions suggests to me that you have not given this enough thought or conducted enough on the ground research to even know if you can possibly make a go of it here - except that you believe you can do it cheaper here than there. I am sure you can make all sorts of changes where you are to save money without having to spend a pile of money to get you, your family and some stuff here only to find out that it's not going to be anything like you envisioned and you in fact gave up far more leaving where you are than could ever expect to reacquire here.
Forget about changing your footprint, get down here, look at real estate prices, rental opportunities, walk through grocery stores, check out the cost of furniture, appliances and the general retail environment in various places. Talk to foreigners asking how much they pay for electricity and if they often have more than one light bulb on at a time. You need to put both of your boots firmly on the ground here after figuring out how much you can afford to spend to see if that is even possible.
Good luck.