OK -- after some more searching, I found that Roeland Van den Eynde created a blog post about this. I have to admit, I found his post a but cryptic, but I was able to puzzle out the steps based on that information.
To copy the custom Themeroller settings, you must have Themeroller open with the settings to be copied active.
1. Open the console of the browser (i.e. in Chrome -- More Tools -> Developer Tools -> Console)
2. Type apex.utr.config() at the console prompt. This will generate an "apex.utr.config" command in the console with a bunch of gobbldeygook settings.
3. Copy the apex.utr.config command into the clipboard.
4. Open the destination app and open Themeroller.
5. Open the browser console.
6. Paste the apex.utr.config command into the console. Themeroller will now have the desired settings.
7. Use 'Save As' in Themeroller to save the settings in the new app.
Frankly this strikes me as something the Apex Development team could have handled better.