You've figured out that playing golf in the DR with a Federation Membership Card is one of the great bargains anywhere in the world. Additionally, you're in luck with the main office being in Santo Domingo you can purchase a membership card and be on your way. that being said, the logistics of your undertaking will render it almost impossible. Here is a link to the membership office with contact numbers. They have English speaking secretaries.
Golf Dominicano
Driving around Santo Domingo in your rent a car and not speaking Spanish it might take you a day to find the place. The memberships are annual so buying in Nov you should get a card for 2008-09. I think the cards are technically for Dominicans or members of a course in the DR so they may ask you for a course affilation. They do not check so you can tell them any course. For about $150 each for the membership you can enjoy a lot of cheap golf especially in the Juan Dolio area with Guavaberry and Metro CC offering tremendous bargains to members ($30 cart included). Some free rounds also. Your best bet would be to call ahead and purchase with a CC. They say can send your card to a member course where you'll be playing but I wouldn't trust them to do that, I'd pick them up at the office. Good luck!