A 3-year-old unvaccinated child from rural Maharashtra presents to the emergency department with acute onset fever, neck stiffness, and altered sensorium. CSF analysis shows pleocytosis (WBC 850/μL, predominantly neutrophils), protein 120 mg/dL, glucose 35 mg/dL (serum glucose 95 mg/dL), and Gram stain reveals pleomorphic Gram-negative coccobacilli. What is the most appropriate immediate next step in management?