We are in California, and we are not required to register ourselves with the insurance carriers, including Medicare and Medicaid. Our doctor clients are required to give certain information about themselves when they apply to become affiliated with a given insurance carrier. The doctors, not us, would be responsible for giving appropriate information - including details about us, if details are requested.
If we were to submit electronic billing directly to the insurance carrier, or to Medicare (which anybody who wants to bother with it can do, if the carrier accepts direct electronic billing), we would need to register with them and provide whatever information they needed - independent of what information the doctor would have to provide in order to be a participating provider.
This is not a legal opinion, but I think that a billing company that submits billing electronically directly to the insurance carrier moves from being a business associate to being a covered entity, as both terms are defined in the HIPAA and ACA regulations. I don't know that you will get two different health-care attorneys to agree on whether I'm correct or not, given that this is new legal territory that is being plowed up and it will take awhile for case law to build up so that lawers are in agreement over the definition of terms.