Car rental is expensive and if you don't plan to be using it every day, I think you would be better off to take a taxi and rent a car just for the periods that you need it.
Have you checked with your hotel to see if it would send a van-type taxi for you? There was another person recently with questions like yours (no dog, though) going to Las Terrenas. He ended up with a taxi sent to the airport by the hotel. He posted a report saying the driver spoke English, was knowledeable about the area, and gave him a wonderful tour on the drive to Las Terrenas. The charge for the taxi will be about the same, whether you take a taxi from airport or a taxi sent by the hotel. Both would have to make the same roundtrip. If I were you, I'd first contact your hotel and ask them for help.
If you take a taxi from airport and vehicles there aren't large enough to comfortably accomodate you and your dog all the way to Las Terrenas, take a taxi from airport to taxi stand in Sosua. There are a lot of van-type taxis there.