I tried it tonight and it worked!
At first I connected them via my multi-meter, and while the light on the pi flashed, it never came up on my wireless network. I observed it pulling between 110 and 170 mA during this time, but the lights on the wireless dongle never came on.
I had decided that there just wasn’t enough current, but took the multi-meter out of the circuit and tried again before giving up. That time (and every other time I have tested) it worked, the pi booted up and joined the network. I’m not sure why my multimeter would keep it from working, maybe I’m not using it right.
Anyway, I’m happy to report that a raspberry eggbot can run with a single power cord.