For short flying times, there is no option other than to use LH via FRA. SQ offers service JFK-FRA, but I don't know if you can combine it with FRA-BOM on LH.
If you are prepared to backtrack you could fly east bound via SIN/BKK using NYC-FRA/MUC-SIN/BKK-BOM, but this will add 6 hrs more flying.
Bombay has plenty to offer if you like big cities. You could take a trip into the interiors to see the countryside or buy a ticket for 200-300$ to Goa and enjoy the beach in good hotels which come for 100$ a night or backpacker places for far less.