Can I travel on other forms of U.S. identification traveling from US to BVI and back?

I am assuming you plan to fly there. While the authorities of the British Virgin Islands will accept a birth certificate and driver's license for entry, the US government requires that you have a Western Hemisphere Travel Initiative-compliant document to travel in or out of the US by air. Unless you happen to have a Trusted Traveler Program card "(NEXUS, SENTRI or FAST)" or fall into an unusual exception (US military traveling on official orders, U.S. Merchant Mariner traveling on official business), that document will be a US passport.

If you can plan ahead, you can obtain a second US passport so you can travel while your first passport is with the French consulate (or pay a passport service to get one for you), but the process is somewhat costly and cumbersome.

I recently traveled by car from Canada to the US and asked US customs officer my question. The answer is yes you can travel from the US to the BVI by air on a Canadian passport, that states you were born in the US.