If a US citizen by parent is involved, then forget the charity aspects for "aliens" and get the US Passport for the kid! No waiting for green cards....you have the go straight to US card....kind of a get out of jail card.
If the US citizen will get the birth registered at the US consulate or embassy, it is just a matter of getting to a hospital inside US territory with the child and his US Passport. If dad needs immediate or "emergency" health insurance for junior, then Medicaid will cover it. And if dad's finances are screwed up from DR hospital bills also don't forget the Children Health Insurance Program for their subsidized health insurance, too. US citizen children are protected and will be taken care of inside the USA. Bottomline, Medicaid will be a social safety net and personal bankruptcy protection is the worst that could ultimately happen to dear old dad. Stop fiddling around with DR hospitals and get the US paperwork filled out for US citizenship by parent (i.e. register foreign birth abroad). Good luck, kid!
