I've been looking at this problem again this evening and I've finally figured out how to get what I want: namely, two themes installed and selectable, the default and one based on it.
There is clearly a major bug with the theme handling process, but the steps below got me around it, so I hope it will help anyone else experiencing similar issues until it gets fixed.
Steps to work around the theme handling issue:
Go to the default theme's folder and edit the config.xml file to set enable/disable on the module lists, so it matches up with the current installation.
Copy the default theme's folder and rename it for the new theme
Edit the new folder's config.xml file to replace the theme name and directory in both nodes – <theme> and <variation> – to the new theme's name and folder
Copy the new folder's config.xml file to somewhere outside the folder.
Go to the back office and install the new theme, but do not try to use it yet.
NOTE: This is where the major issue occurs: it states the upload has been successful, but if you go to the new folder it has been almost emptied out! Just the docs folder remains.
Re-copy the contents back into the new theme's folder and copy the config file back in.
Now you can choose 'Use this theme' in the back office and it should work.
Needless to say, as the settings in these two themes diverge it will be prudent to copy both folders and edit their config.xml files as necessary before switching between them.