Have you tried a full recharge of the charger overnight, then repeated again? Sometimes these high capacity LiIon batteries take a few charge/discharge cycles to work properly.
If it still cuts out during a charge of the HD2, then won't restart it suggests a hardware fault- maybe a duff cell? That would need a replacement.
I have the larger 13000mAh version and it has a similar fault, though it can charge my HD2 about 5 times before it falls over. Mine's out of warranty though
