"We managed to overpower the invaders. but not before
they took out the main reactor." "Battery power is about to die." "We're dead in space. with
twenty minutes of life support left."
This doesn't sound like they where able to restore battery power, for me it sounds more like the batter where ok but running out of juice.
I'd assume that they were able to restore battery power, but not life support, after their final log entry and before meeting their demise.
It's also apparent that Rebecca boarded the derelict at some point in 3273. Maybe she managed to power up the reactors using spare parts stored aboard her Cobra.