It doesn't exist. Every hard case I've seen has a fairly standard width of 1.9". Second, Lipo technology hasn't advanced to the power density you're looking for. The best capacity you could expect from a battery that size is around 2200mah. Something like this Turnigy 2250mah softpack which is 4" x 1.38" x .75"
https://hobbyking.com/en_us/turnigy-...lipo-pack.html