Boosting Peak Draw spec on LiFePO4 using Capacitors
Without knowing the characteristics of either the LiPo protection circuit or the peak load it is difficult to give a detailed answer, but my first (very confident) response is "No." You have fundamentally misunderstood the relative natures of batteries vs capacitors.

A battery is (to a first approximation) a constant-voltage device, and a capacitor is not remotely like it. Your calculations, for instance, show that a 0.8 F capacitor being discharged at 12 amps will drop from 15 to zero volts in one second. After, let's say, 0.1 second the capacitor's output at this current will have dropped from 15 to 12 volts, and at this level I'd expect the battery to be providing all the current, so the protection circuit will kick in. Making a capacitor supplement as you wish requires a detailed knowledge both of the load drain profile and the protection circuit's behavior. Basically, you'd expect the battery voltage at maximum current to drop to some lower (but non-zero) voltage, and the capacitor would take up the slack - briefly. However, as soon as the capacitor has discharged to the nominal loaded battery voltage it will not contribute ANY current and the battery will have to supply all the current. And we know how that works out.

In principle you could use a very large capacitor, but you would have to determine how the protection circuit works. Let's say that protection circuit reaches its trip point at 12 volts output, with a constant 22 amps output from 15 to 12 volts. Then the capacitor must provide all the energy of the flash by dropping from 15 to 12 volts. Then, since the load needs 600 J and the formula for a capacitor's energy is $$E = 
\frac{CV^2}{2} $$ we can say that $$E(total) = 600 = \frac{C}{2}(15^2 - 12^2) $$ and $$ C = \frac{1200}{81}= 15F$$ and since the recharge rate is $$\frac{\Delta V}{\Delta t}= \frac{i}{C}=\frac{22}{15}=1.5 V/sec $$ the recharge time is $$T = \frac{3 \text{ volts}}{1.5 v/sec} = 2 \text{ seconds}$$ which is close to your desired cycle time. Of course, assuming constant current capacitor charging is probably not a great idea.