The Droid Sans Fallback font is much smaller (unpacked deb is about 7 MB) but still allows to see all the characters on that sakura site. It's available from the PPM on Fossapup, maybe on Xenial too. Try to search for fonts-droid-fallback.
It's not clear why the browser can't use your fonts. On Fossa only dillo refuses to see the available fonts and shows squares instead. If you can see the Japanese text when you copy the squares in the browser and paste them to the text editor, then something is wrong with the browser settings. Maybe it is configured to use a specific font all the time. But firefox shows unicode chars regardless the settings.