This rings a bell.
See the
endless saga of 'el ilegal' and
the update.
What most dual nationals do is to leave the country on their other (in this case EU) passport, and enter with their DR passport.
However, as Piblet was born in the DR, and it should state this on her Danish passport, she does not have to enter the country as a tourist, even if she doesn't have a DR passport. I travelled to Haiti with a Dominican colleague who holds a EU passport, and when we arrived back in the DR he didn't bother showing his DR passport, he showed the EU one. The immigration chappy said 'tourist card?' and he said, overdoing the Dominican accent - 'soy dominicano, hombre, que vaina?' and showed him the bit where it said - 'Place of birth: Santo Domingo'. Maybe you need to train La Piblette in something on these lines.
But the best and easiest thing would be to get her a DR passport - we're finally getting one for Chirimoyito after years of tourist cards/penalty payments. Last year it was RD$2,000 for 9 months. This year with the DR passport it will be RD$0. In your case it will be straightforward because she was born here. We had to go through a more complicated legal process because he was born outside the DR to a Dominican father in a place where there was no DR embassy to register him in the requisite 6 month period.
Based on what I've been told by Fabio and others here, a minor cannot have residency anyway.