If I was flying, I'd take BNA-anywhere KL or AF flies-CDG or AMS-BOM. You want to maximize your time on the foreign carrier (and not just because of the free alcohol!)
ie, BNA-DL-ATL-AF-CDG-AF-BOM, or BNA-NW-DTW-AF-CDG-AF-BOM, or BNA-CO-EWR-AF-CDG-AF-BOM