Link to referenced FAQ Page
This part of the FAQ suggests that the APE, MPC... option also works with FLAC files. Based on my problem above, it does not. I don't know if there are any official rules about what tag types are allowed on certain file types but I have heard of other people with ID3 tags on FLAC files so it can be done.
I was able to solve my problem. I used MP3Tag to cut the Vorbis tag off to the clipboard, then a different (gasp!) tagger to cut the ID3 tags off, and then MP3Tag to paste the Vorbis tags back on. Worked fine, if not slowly and time consuming. I still think MP3Tag should have an options page just for FLAC and its tags, as more and more people are using FLAC for archiving.