I understand and sympathize with you situation a birth certificate doesn't prove anything just like you described. But the USVI is not the US and is not US soil. It is a territory with its own laws, government, and customs. They drive on the left side of the road and can't vote in federal elections. I love the USVI but when I go there I don't have the attitude that I am on US soil.
My best advice is any time you board a plane that will be landing outside the continental united states , know what documentation you will need to return. And never expect anyone one from customs to be friendly or helpful especially in the usvi. Some agents are great but others are very unpleasant to deal with.