We have designed an led lamp that is run by a constant current source. The constant current source requires a minimum of 6.2 volts. With a 9 volt input to the constant current source the current draw is 400 ma. As the voltage input drops (battery voltage) the current will go up (constant current = constant power). We need a rechargeable battery and charger for this application. We would like the battery life to over 1/2 half hour and in a small package. What is a good choice for this application?