Hi there,
I would suggest that you take the time and the little extra effort to go to Samana. I myself, don't know of any "private" and secluded hotels in Bavaro. Well, someone here mentioned last year a B&B near Bavaro called Cortecito Inn, but that is all. There are secluded beaches in Punta Cana area for sure, you will just have to take a tent and camp there
As for Samana, I think this place definitely offers what you're looking for. You can find small inns and hotels for very reasonable prices. There are tons of hotels to choose from. If you need more information, why don't you send Andy an e-mail. He is the owner of El Marinique Resort in Las Galeras, Samana, and a regular contributor on this board.
The following are just a small list of options in Samana:
El Marinique Resort
Villa Serena
Club Bonito
El Portillo Beach Resort
Gran Bahia Hotel
Casa Marina Bay
Also, try to look at debbiesdominicantravel.com for extensive reviews of resorts in the DR by independet travellers.
Hope this helps,
Natasha