So I don't do flywheels, but I know a few things about electronics and feel I need to explain some things.

 

The capacitor you need to do this probably costs more than the bigger battery, without giving you any increase in capacity that you would get with a battery.  This is kind of a weird edge case where it's probably not absurdly expensive, but still more expensive than just using a bigger battery. 

Now, we have ready access to supercapacitors, which are still fairly inexpensive, but can deliver a ton of current.

Here's an example of a website selling them, as you can see, they have rather high energy density and fairly low pricing.

https://www.richards...=Ultracapacitor is a retailler local to me, and judging from the specs, you would need a small array to boost voltage to 14.4 V (3S) but it seems to check out.