Is there a medical team or institution that is uniquely specialized in working out complicated diagnoses unfamiliar to most doctors that involve endocrinology, rheumatology, and possibly other areas?

Endocrine: Most university hospitals have endocrine departments. Some well known ones are mayo rochester, hopkins, but there are many others.

Teaching institution: Yes teaching institutions such as the mayo clinics are a great place to start.

Mayo Clinic: I would use mayo clinic. They have campuses in minnesota, florida, and arizona.

Academic center: You may be looking for an academic center. UT Southwestern has good specialty departments.