Burn Pit Registry Enhancement Act This bill authorizes specified individuals to update the Airborne Hazards and Open Burn Pit Registry with the cause of death of a registered individual. The Department of Veterans Affairs established the burn pit registry to allow certain veterans and service members to document their exposures and report health concerns. The registry is for individuals who may have been exposed to toxic airborne chemicals and fumes caused by open burn pits (areas used for burning solid waste in open air without equipment) while deployed in certain locations (e.g., Afghanistan or Iraq). Specifically, the bill authorizes a designee of a registered individual or an immediate family member to update the registry with the cause of death of the individual.