Final answer:
The most appropriate initial step in the treatment is to obtain an 4) ophthalmology consultation due to the serious nature of the child's symptoms, which may indicate an orbital infection or another serious condition.
Step-by-step explanation:
The most appropriate initial step in the treatment of a child presenting with fever, lethargy, proptosis, limitation of ocular motility, and pain on eye movement is to obtain an ophthalmology consultation. These symptoms are indicative of a serious eye condition that requires immediate specialist assessment. Possible underlying conditions could range from an orbital infection, like orbital cellulitis, which would require urgent antibiotic treatment, to more serious conditions such as a space-occupying lesion causing increased intraorbital pressure. An ophthalmologist is best equipped to evaluate these symptoms and facilitate the necessary diagnostic imaging, such as MRI or CT scans, which can guide further management.