OK, I did something dumb. I followed Apple's instructions for removing songs from my pod by deleting the playlist in iTunes.

When I connected the pod, I got the error that it couldn't update because the smart playlist was gone.

I added a new smart playlist with the same name as the one on the pod, then tried again.

This time, I got the error "The iPod DeePod cannot be updated because there is not enough free space to hold all the songs in the selected playlists"

I checked the playlists, and they contain far fewer songs than the 1017 that are currently on the pod.

Any ideas what to try next?