Tokyo to Mt Fuji to Kyoto – I may have made a mistake booking Lake Kawaguchiko. Better to do day trip to Mt Fuji from Tokyo vs stopping halfway to Kyoto?

First time in Japan, family of 4. Spending 4 days in Tokyo, 1 day in Mt Fuji (Lake Kawaguchiko), then 4 days in Kyoto. Was thinking we would spend the night at Mt Fuji.

Now I’m learning that with the shinkansen it’s much faster to just go from Tokyo to Kyoto direct – 2+ hours. Whereas if I stick to stopping at Mt Fuji one night, we’ll be towing the kids on buses and trains for half a day each day as we travel between Tokyo-Fuji-Kyoto.

Does it make more sense to just spend another night in Tokyo, make a day trip to Hakone / Mt Fuji, come back, then go straight to Kyoto?

Thanks for the insight.

by OtherwiseBase5003