By "non excursion" do you mean an independent trip by car or public transport?
You can go to Isla Saona independently and negotiate a fee with the local boat owners from Bayahibe, which is only about an hour and a half drive from B?varo. You can go to Boca de Yuma, a fishing village to eat seafood and visit the Ponce de Le?n house/museum in San Rafael de Yuma on the way - also just over one hour by car. It's also possible to cross to Isla Saona from Boca de Yuma with a local boat owner. Otherwise a day out at Macao Beach, about 20 mins by car. You can also visit the city of Higuey and its basilica (45 mins drive).
Other options are the Ojos ind?genas reserve (freshwater lagoons in a rainforest setting) in the Punta Cana Resort and Club, the newly inaugurated Laguna de B?varo reserve in the B?varo-Punta Cana area.