A 5-year-old girl from Mumbai presents with a 3-day history of fever, headache, and vomiting. On day 3, she develops a non-blanching petechial rash over the trunk and extremities. Her parents report she attends a crowded school. On examination, she is febrile (38.8°C), alert, with neck stiffness and a positive Kernig's sign. CSF analysis shows: protein 150 mg/dL, glucose 65 mg/dL (serum glucose 110 mg/dL), WBC 200/μL (60% neutrophils, 40% lymphocytes), Gram stain is negative, culture pending. Blood culture grows gram-negative diplococci. What is the most likely causative organism and the immediate management priority?