I don’t think BrowserFrame.addStyleName will work, as that probably adds the stylename to the IFrame itself (or more likely an element wrapping the IFram).
If you don’t want editors of the documents dealing with the styles, you could also do some preprocessing to the contents before you embed it and put them style block in the file in code.
HTML imports are indeed not usable in all browsers without extra polyfills: https://caniuse.com/#feat=imports
-Olli