The car is a 2005 Toyota Corolla CE. Its engine is the 1ZZ-FE, a 1.8L straight four that runs on gas, not diesel.
The alternator is stock, and tests out just fine on the multimeter in both voltage and amperage. However the battery charge is very low. After trickle charging it overnight, and starting the engine on the battery booster, the car drives a short while, then the engine stops.
My hunch is a disconnect in the electrical system, like a corroded wire. However, I would like to know if it's possible for a bad battery to cause the engine to stop. My understanding is that the battery is used only for starting the engine, but that may be overly simplistic.