Thanks Oliver for your efforts! A professional affiliated with the hospital searched flights all over the country and found EgyptAir offering business class flights between Bombay and JFK (New York) for as low as $2,189 INCLUDING TAX (Round trip, business class)! For this price, I think we could get one more kid in for surgery.

Orbitz and the other engines apparently don't throw up this option:
EgyptAir 969 from Bom to CAI and 985 from CAI to JFK
EgyptAir 986 from JFK to CAI and 968 from CAI to Bom.

I don't think any other airline can beat $2,189 round trip inclusive of all taxes for business class, so my search has ended. Thanks to Oliver and everyone on this forum.

Maybe I'll use this forum for myself henceforth since I am always traveling!