In addition, the usual escape route (command-option-escape) wasn't functional. The dialog box would not show up on the screen, regardless of which app was in the foreground.
I could have used another computer to connect and quit the Finder via "ps aux" and "kill", but that seemed like cheating. The only app I had in my dock (the only way I had to start programs) that wasn't running was CPU Monitor. On a lark, I launched it and noticed that it has two very useful menu items for troubleshooting system lockups - under the Processes menu, you can choose "Open Process Viewer" or "Open Top".
Either one of these was enough to solve my problem -- opening Top launches the Terminal, from where I could open a new window and use "ps aux" and "kill" to relaunch the Finder, or I could (as I did) use ProcessViewer to do it directly.
The moral of the story? To prevent a Finder lockup from rendering your machine unusable, keep one of CPU Monitor, Terminal, or ProcessViewer in your dock at all times. My personal choice is CPU Monitor, since I can get to either of the other apps through CPU Monitor. With one of these apps in your dock, you should (barring a dock lockup!) be able to launch a program to help you restart the Finder.
The nice aspect of working in OS X, of course, was that everything was back to completely normal once I restarted the Finder -- and, much to my amazement, I found that the copy that had locked the Finder had actually completed!

