While I get your drift of NOT using the warhead in high population density first (honourable an idea), I think that probably would be filed in World History under "Well intentioned but out of Depth people causing unintentional world catastrophe" if went ahead. (disclaimer: Not Geologist).
Mount Fuji is still considered an ACTIVE VOLCANO, even if less so this recent centuries. If you scrap the multi square kilometres of cover on Mount Fuji with a sudden, catastrophic blast you'd probably can trigger a geologically unprecedented eruption. That probably has good likelihood to cause massive or total destruction in level of at least 10,000s of Square Km, spanning multiple Prefectures of Japan, and the death & destruction probably would dwarf both the Atomic bombings by order(s?) of magnitude.
Not to mention the aftermath of the unprecedented volcanic+atomic blast on a 3776m high Volcano have a good chance to cause the whole world to go into a Impact winter, that have a chance to top the Mass Extinction event 66 Mil years ago. And even if not, the carrying of NUCLEAR volcanic ashes spanning way across China and Russia likely would expose 9-digit-strong population to radiation.