Don't think of it as the "header," since the bext chunk, like any other chunk, can be located at any point in the file. I've seen plenty of .wav files that have the bext chunk after the data chunk.
Getting to the bext chunk simply requires parsing the file chunk after chunk (like you're probably already doing for the fmt and data chunks) until you get to ...
I'll answer this myself. The problem was a structure alignment issue that threw off the positions of some internal fields. Once that was resolved, the files work correctly.
I never needed metadata on WAV files before, but I can see how useful it'll be.