Copyright is granted upon fixing your original work in a tangible medium of expression. It applies to published and unpublished works alike. You do not have to register anything. No one else even has to know your work exists. Once it is fixed in a tangible medium of expression, it is protected and you retain the rights until you voluntarily give them away or until the work becomes part of the public domain.
What you would be doing at copyright.gov or BMI is registering your song. The purpose of registration is not to copyright your work; it is to create a public record of your creation so that you can prove your work came first. With no such record, anyone who infringes on your copyright could claim you copied them. For this specific end, the BMI registration would be just as effective.
In order to bring a lawsuit in the US, however, you must register your work at copyright.gov. As registration is strong evidence in court (especially when done prior to or within five years of publication), it is highly recommended that you do so if infringement is likely. Many people simply register every serious work they create in order to be as secure as possible. This is probably unnecessary, but better safe than sorry.
Disclaimer: I am not a lawyer. The above is based on seminars I’ve taken on copyright law, discussions with colleagues who have studied and/or practiced copyright law, and FAQs available at copyright.gov. This information may be out of date or based on a misunderstanding. It is always advisable to speak to a practicing lawyer.