A good way to storage/precharge energy for short bursts of consumption?
2 amps for 2 seconds at 6V is 24 joules of energy and if this was provided by a capacitor the capacitance should be at least 5 times that required to meet the energy equation. 

Energy in a capacitor is \$\dfrac{C\cdot V^2}{2}\$ therefore \$\dfrac{2 \cdot 24}{6^2} = C\$

C is therefore 1.333 farads but realistically it needs to be 5 times that so it doesn't become totally depleted during the time when the energy is needed in the 2 second burst. More like 7 F is needed. Make sure it's a low ESR type!!!

A problem occurs when the the requirement to drive the heavy load occurs too often. You said in your question that there might be a 10 second gap between "usages" and I infer only 200mA is available for restoring charge in the capacitor. 200mA at 6V for ten seconds is an energy availability of 12 joules and the only way around this is to have the capacitor as large as possible so that even if two "big demand" periods occur over a ten second period there is still enough residual change in the capacitor to service both requirements. Maybe go for a 14 F cap.

Can you get these?