Sounds like a fun trip. I would suggest you spend a bit longer in each place if you can. Maybe you’ll feel a bit rushed doing a day in each place :/
I think the JR Pass is super useful, especially if you’ll be using the JR in Tokyo as well (this is mainly the Yamanote line around the centre of Tokyo).
For seeing Mt Fuji, have you checked out Lake Yamanako? I think this is one of the most incredible views of Fuji you can get. My friend was in Hakone last weekend and didn’t get to see Mount Fuji because it was too sunny. If you’re set on getting a good sighting, I would suggest getting up early in the morning wherever you are as it’s normally clear early in the morning.
Hope that helps,