I can't recommend any service.
I took a look at klook and saw some options to Fujikawaguchiko area and Hakone. 5-hours would be $340 to $440 per car for up to 6 pax. They have some fixed routes but also a custom option. The say pickup in Tokyo city which I assume includes Shinagawa.
Rome2rio says about 100 minutes' drive each way. Your timing is good, getting to a Fuji view maybe before clouds pop up.
I would not want to do this on a weekend, especially a Friday, because of traffic. OTOH, you might get a jump on the weekend get-awayers and maybe see some backup on the road going in while you are on your way out at noon.
300 min tour - 200 min drive = 100 min at destination.
Head to Lake Ashi in Hakone near the pirate ship pier. Great views, hopefully, of Fuji-san across the lake. There is/was a restaurant. Have some time to walk a bit of Tokaido Road among some cool trees.

I was there many years ago, and maybe it is crazy in Hakone these days like it was Kawaguchiko 2 or 3 years ago.