A 42-year-old woman with rheumatoid arthritis on immunosuppressive therapy presents with acute swelling of the left wrist and hand joints. She has fever (38.8°C) and elevated inflammatory markers. Arthrocentesis is performed and synovial fluid shows WBC 45,000/μL with 85% neutrophils, but Gram stain is negative and initial culture is pending. What is the most appropriate next investigation to identify fastidious organisms and guide early therapy?