https://www.fujisan-climb.jp/en/Regulation_Shizuoka.html

I think not pre-registering makes the most sense to me, cause if you pay in advance, you cannot cancel or refund the payment - nor change the date after confirmation. And since I got screwed over previously with a typhoon on the day I wanted to climb, I'm not willing to risk having a bulletproof itinerary on the weather, or any other outside factor.