Also - are you sure you’re not thinking about visiting the Fuji 5 Lakes area when you say you want to “explore Mt. Fuji”?
I don’t actually know what that means with relation to a Hakone visit. Suggests that you expect to do more than just see Fuji San (which there is no guarantee of in the Hakone area).
To be honest, I’m not actually sure what “explore Mt Fuji” actually means in April as you can’t climb it then. “Visiting Mt Fuji” keeps coming up as a fixture in people’s Spring and Autumn plans but I never see any reports back that explain enough for me to grasp the concept. Would love to find out finally!