With a limited permit from New York State to practice medicine as an international medical graduate, Dr. Bello was essentially an extra pair of hands for the department of family medicine at the 17-story hospital, one of the biggest and busiest in New York City. Dr. Sridhar Chilimuri, the hospital’s physician in chief, said Dr. Bello could treat patients and prescribe medication, as long as other doctors were looking over his shoulder, and only at Bronx-Lebanon. “Not over there — not in a clinic,” he said for emphasis, pointing out the hospital’s doors. But Dr. Bello’s slow journey to the...