That worked. Thanks!

The secret to migrating to another machine seems to be to switch to INI mode, copy the INI file, switch back to registry mode, delete the INI file.

WinSCP by default stored configuration to Windows registry, so you do not have these kinds of problems. It uses INI file only if you force it to do so, like by copying INI file to its installation folder. What you can do is to start WinSCP (let it load the INI file) and go to Preferences and switch to registry storage. It would fail when it tries to remove the INI file, but otherwise it should copy the settings. Then quit WinSCP and as administrator, remove the INI file.
See
https://winscp.net/eng/docs/ui_pref_storage
https://winscp.net/eng/docs/config

I used the installer version of WinSCP 5.1.3 on my Windows 7 (64-bit) box. Unless I run WinSCP as Administrator, it is unable to save changes to the configuration. But if I run it as Administrator, it will also run my external editor as Administrator, which breaks Explorer drag-and-drop. I can't disable UAC (nor really want to) because this is a machine in a corporate environment and IT would shoot me if I disabled it.

The problem is that WinSCP wants to store its configuration in 'Program Files (x86)\WinSCP' instead of the user directory. I suspect SSH keys are stored the same way because, every time I (re)connect to a new server while not running WinSCP as Administrator, I get a dialog asking if I want to allow the SSH key.

I should be able to run WinSCP and make configuration changes without running it as Administrator.

The configuration was imported from a WinXP environment, which may have caused this problem, but I'm not sure how to fix it.