Yeah, a computer power supply will not maintain a solid 12v rail, and as the voltage drops, the charger's built-in power supply will try to draw more current to compensate. When converting PC power supplies, I always try to boost the 12v line as close to ~14.5v as possible.
This thread talks about the various ways this may be achieved, depending on the design.
And,
this page will help you figure out the power supply you'll need. Since you're only looking for a supply that will charge that specific pack, don't use the drop-down list to select your charger (if it is even in there); just enter your data in the fields.