I don't think the insurances companies really care who pays the health insurance premiums. Getting insurance for the child and not the mother is going to make her brain spin and probably convince her to take you to the cleaners for child support and some of her living expenses if you aren't living together.
If you want to provide "the best" for your child, you'll get him/her out of the DR at the first opportunity assuming the mother consents. I don't think a very young child needs US citizenship right away but if you are inclined to do that, doing so sooner as opposed to later may be better. If anything happens to you that application becomes more difficult. If the child frequently travels outside the DR a US passport will be invaluable.
Unless you plan to never look back, you should probably get your name on the birth papers soon so that you can exert your parental rights if need be. It takes more than 1 at home parent to excel at child rearing.
You're right, it's cheaper to raise children here but look at what you get for your money.
Good luck. The north coast could use a McDonald's if you decide to give up the yachts and put down roots...