Now that I had them in Font Book, I could remove all the duplicates by selecting all the fonts with dots by their name and choosing, File » Remove "Font name." Things were looking better, but I wanted to install these fonts on another computer. So I exported them by using Edit » Select All, then File » Export.
To my surprise, Font Book created subfolders by font name for every font. Nice and organized, free from duplicates. As a final check, I ran File » Validate File and chose my new 'Exported Fonts' folder. That found a couple more issues with some suitcases that I fixed by following this thread on Apple's discussion board. When I was done, I had a clean set of fonts ready to distribute.
[robg adds: I don't know if this one also works in 10.4; for now, I've not tagged it '10.5 only,' but if someone can test on 10.4, please add a comment.]

