It does work USB powered. I have tried a a spare LiPo batter (much smaller mAh) that I have, and connecting this gives the red charge LED. Therefore, I think the battery is faulty. Can a replacement be shipped?
Partially related - is it possible to put two batteries in parallel to double capacity. I'm a aware generally this would not be a good idea, but don't these small LiPos have on-board charge circuits?