Well, I managed to figure it out enough that I was able to get it working.
Tried deactivating GPP and replacing it with the new version. I would have had to delete the old and add the new, which (according to the dialog box that came up) would have deleted all of GPP’s data. Oops. Nope.
Tried downloading the GPP ZIP archive, uploading it to the plug-ins folder, deleting the old GPP version folder, and unZIPping the new version there, as I would do under localhost. My FTP client wouldn’t unarchive the ZIP file.
Finally, I went back into the plug-ins page, added the ZIP archive, and reactivated it, then went back in via FTP and deleted the ZIP file that didn’t get unarchived.
At least this way I didn’t lose my settings.
I was able to reauth GPP and all seems to be well.
I do note that sometimes, GPP doesn’t seem to “know” that updates are available, even on sites running most-recent versions.