Wow
I hope you saved a lot of $$ by flying into Santiago instead of Puerto Plata, because you are putting yourself in what IMHO is an uncomfortable position with a child in tow.
There will be cabs waiting, and a cab ride to a hotel should be no more than RD300 each way. Another problem with arriving at that hour is the need for pesos. I don't know if there's an ATM machine at the airport, but if not the cab can take you to one. There will be noplace open to exchange money.
I would still say that a cabana would be the most convenient and cost effeceint place to grab some sleep. There are a lot of them near the airport on the highway, and because they are rented for short periods of time they should be inexpensive in your situation. Here's the problem; and only you can decide if you are comfortable with it, having your daughter with you. Cabanas are "love hotels", designed to accomodate Dominican men for discreet romantic interludes with their mistresses. Security should be no more an issue than in any hotel, the rooms are nice, and very private. However, considering their purpose, I don't know that they will have any rooms with 2 beds. The one that is there should be at least queen size, if it isn't one of the big round ones. In addition, you will need to be careful with the TV, since it will no doubt be tuned to a porno channel, although other options will be available.
If you aren't comfortable with this, you will need to go a bit further and pay more for a "regular" hotel room.
Having said all that, my personal recommendation would be to contact the transportation provider recommended in your other thread and have them take you straight from the airport to your accomodations in Sosua. If you then want to rent a car, you have plenty of options. In addition, you may find you don't need one once you get there. Sosua is very "compact", and many people get around fine without a car. One of the most prolific posters on this board has lived in Sosua without a car (by choice) for many years.