[rthinman]

You can get a quick estimate of whether those 4000 uF and diode will be sufficient or a waste of time. Assume:
The LEDs draw constant current I (they won't, that's why they dim)
The battery drops out for a time deltaT, perhaps 2 seconds, while the shooter spins up. For that entire time, it is lower than the capacitor voltage so the diode is off.
We can only tolerate a certain amount of drop deltaV in the LED supply voltage before our constant current assumption is now longer good, or that the LEDs are too dim. Maybe use 1 Volt for deltaV.
Now since we've made assumptions of constant current, we ignore the calculus and estimate:

I = C * deltaV/deltaT ==> C = I * deltaT/deltaV

With the above assumptions and I = 200 mA, we get C = 0.4 F = 400,000 uF.
You will probably want C much bigger than that to cover our assumptions.

So Al is right, ordinary capacitors generally won't cut it. Super capacitors (also known as ultracapacitors and "gold" capacitors) would be a possibility, but they come with other problems. You will want to put a small-value resistor of sufficient wattage rating in series with the diode to temper the inrush current when you turn on the Spike.

Rod