Fixed it. I needed to add the com.apple.ibooks.display-options file to the package. Looks like this:
<?xml version="1.0" encoding="UTF-8"?>
<display_options>
<platform name="*">
<option name="specified-fonts">true</option>
</platform>
</display_options>
And yeah, there's a lot going on in those h1 tags, but the formatting is pretty complicated. Looks good when it actually works properly.