As he was born in the DR even with American citizenship and passport, he needs permit from the mother. Follow the link at DGM website.
Mother needs to sign permiso de menor, needs to be notarized, and legalized at PGR. Then you need a certified copy of the birth certificate. Upload the whole package to DGM system and then go pick up the legalized permit. If child will be traveling alone the father needs to do the same thing on his side. Your costs will be paying notary, paying for birth certificate, paying for PGR legalization, and paying for DGM permit. Total cost about 5000 pesos. Does not include fees if father needs to do the same thing, in such case he must legalize his permit at the DR consulate or with any local notary plus Apostille, then shipping the permit back to DR.
And no, dollars will not grease the palms at the airport If he tries to fly without the legalized Permiso. It used to be until last August that children with double citizenship traveling on foreign passport didn't need permit. This is no longer valid and now they do, if they stayed in the DR longer than 90 days from their entry. Especially if their foreign passport states place of birth DR.