I've heard of that company before. How did you end up buying one of their laptops?
I tried running my celeron M based laptop on a 12V battery, and no go. Three 6V tractor batteries in series for 18V might work to keep it running for a few days.
Does the external pack shut off its output permanently when you plug in it's charger? Maybe the charger can only handle the load of recharging the pack, and nothing more, so they did this to protect it. I suppose you *could* connect the original laptop power supply through a blocking diode, and plug the p/s into the wall through a UPS.
Or get 18V of lead/acid storage, use the 19(?)V laptop p/s as a charger, and connect that battery through a diode to the computer. Fourteen nickel-m.h. cells in series might work with that p/s, without blowing up, if you work out a good circuit.