There is no national registry or licensing for businesses or business names in the US.
DBA is a local issue. It is a power delegated to the states, and, in most cases, further delegated to county or city governments. In most places, it involves submitting an application to a local government, and then publishing a notice in a "newspaper of record". But laws vary widely.
I think your best bet is to contact an attorney who specializes in U.S. business formation for foreign nationals. He can discuss the advantages/disadvantages of incorporation, LLC, etc. and/or options for representation.
We do have something called a Federal Employer ID Number (EIN) number, which can, in most cases, be substituted for a SSN. Not sure what the rules are, but pretty sure any U.S. corporation can get one. It was originally intended for employers to use in making filings related to their employees (such as submission of withholding taxes), but has taken-on a number of additional uses. It is used in many cases as a substitute for an SSN for any entity other than a natural person. For example, I obtained an EIN because I have a retirement trust. The trust isn't a person, and so can't have an SSN. The EIN is used instead.