I don't have encyclopaedic knowledge of Hakone's attractions nor as much motivation to look things up as you do. Wednesdays (and Thursdays) ring alarm bells for me, hence the suggestion that you check.

Seeing Fuji San was a priority on my own visit to Hakone. I went straight out to the rope ways as soon as I had left my bags at the lodging I was staying at as there was some visibility on the day of my arrival. Glad I did as Mt Fuji couldn't be seen the next day.