The only thing that would bring back the electrical was disconnecting the battery cables and reseating them. Every single time.
Using that as a clue, I inspected the cables more carefully. Both clamps were a little loose and had some play. I took them apart, sanded/brushed the wires, scraped away corrosion, and even sanded the ground to the engine block. I reassembled everything tightly, and it's started every single time since.