A 5-year-old unvaccinated girl from Uttar Pradesh presents with a 4-day history of sore throat, low-grade fever (38.2°C), and difficulty swallowing. On examination, she has a grayish-white pseudomembrane covering the tonsils and extending to the pharynx, with significant cervical lymphadenopathy and mild neck edema. The child appears lethargic. A throat swab is sent for culture. While awaiting culture results, which is the most appropriate immediate management?