When the power goes out all over the neighborhood of the batcave, the batcomputer quits functioning, but lights on some of the electronics in the background are still flashing. They're not likely to be battery powered or connected to an external generator because the batcomputer would likely get its electricity the same way as well.
Pinky Pinkston convinces Commissioner Gordon and Chief O'Hara that Britt Reid is Batman and Bruce Wayne is The Green Hornet. This is despite the fact that both Reid and Batman met with Gordon and O'Hara at the same time in Part 1 of the story, the previous episode "A Piece of the Action."