Got 13 days in Japan — cut a day in Kyoto or Tokyo for Mount Fuji?

Hello everyone!

We’re planning a 13-day trip to Japan at the end of October / early November this year. Our main stops will be:

Osaka: 3 days (2 and a half, really)

Kyoto: 4 days

Tokyo: 5 full days (we depart on the morning of day 13)

My boyfriend wants to add a day trip to Lake Kawaguchiko to see Mount Fuji.

I personally prefer to spend more time exploring Tokyo.

Would you recommend fitting Kawaguchiko in as a day trip from Tokyo? Also, would it make sense to reduce one day from either Tokyo or Kyoto to make room for it?

Any advice on how to best organize this would be much appreciated!