What you have for purposes of Privacy and FCRA is an affiliate. What you have for purposes of RESPA is an Affiliated Business Arrangement, triggering the required notice before the referral is made. In fact, it should be given to the consumer before taking the application. And yes, the affiliate is a mortgage broker. A mortgage broker is an entity that brings together a borrower and a lender.
You also have fair lending issues -- steering and prescreening. Because of that, you should have clear, detailed procedures to determine how customers get referred to different lenders in a way that is not discriminatory or predatory.