Osaka to Tokyo - day at Mt Fuji?

Hello,

On the way to Tokyo from Osaka we would like to spend the day at Mount Fuji (we are traveling mid November).
I am checking the different forums (and Hyperdia, Google Maps, etc.) but I actually would need some advice :)

I was looking to go to lake Kawaguchi since, from what I've read, it looks like one of the best places to see Mount Fuji during fall. I've read in a forum that you can take the shinkasen from Osaka to Mishima Station and then get a bus to Kawaguchiko, but when I look in Hyperdia I don't find any route that has less than 5 transfers.

So, long story short: what is the best way to go from Osaka to Kawaguchiko? Or does it make sense to go to another city/lake to see Mt Fuji?

Thank you very much in advance

but when I look in Hyperdia I don't find any route that has less than 5 transfers.

That's because Hyperdia does not give bus connections.

The timetable for buses between Mishima and Kawaguchiko can be found below. It is indeed the best way to travel between Osaka and Kawaguchiko:
https://www.japan-guide.com/bus/gotemba.html

From Kawaguchiko back to Tokyo, a direct bus or travel by trains is easiest. See below:
https://www.japan-guide.com/e/e6905.html

Leave Osaka early. The sun sets in November at the 4 o�fclock hour.

Thank you very much to both @Uji and @John B for the useful information :)

Mt. Fuji is not "on the way" to Tokyo. You are going to be 4.5-5 hours of travel to get there from Osaka, which means you had better catch the first train at 6Am if you want any meaningful time in the 5 lakes area. And you'll have your luggage, which means finding lockers in Fuji area and lugging them through multiple transfers.

The Mishima bus info is here: http://bus-en.fujikyu.co.jp/highway/detail/id/29
It can be the shortest route, but most of the transfers are timed so that you'll be waiting for almost an hour for the bus. Also, there are only a couple HIkari bullet trains a day that stop there and the rest of the day it is Kodama, which adds an extra hour to the trip. So the end result is that for most time intervals, it ends up being no faster, and less convenient, than just taking a faster Nozomi all the way to Yokohama or Tokyo first and then transferring to local lines for Fuji.

I would probably just say head straight to Tokyo and then do Mt Fuji as a day trip from there on another day. You can schedule an early bus from Tokyo, Shibuya or Shinjuku station and be there easily by 9 or 10 am for only about 2000 yen each way. That gives you a much longer time in the area and less stressful day. You can also leave your luggage in the hotel.

Now if you wanted to add a night in the Mt Fuji area, that would be a different story.

"Or does it make sense to go to another city/lake to see Mt Fuji?"

On that note, another itinerary I can think of is, you can visit Lake Tanuki and Shiraito falls on the way to Tokyo. We did the same while traveling from Kyoto to Tokyo in 2017. It is little time consuming to get to Lake Tanuki (from Osaka), but very much doable. An early start from Osaka would help.
Take Shinakansen from Shin-Osaka to Shizuoka, then transfer to local train to Fujinomiya (you can check this part in Hyperdia). There are buses that depart from Fujinomiya station to Shiraito falls and Lake Tanuki and back to Fujinomiya station.