Welcome to Flyertalk, Joshua! I hope one of the ryokan experts has a suggestion, but I'd like to recommend that you consider spending that night in the Hakone area. It's a simple destination to get to from Tokyo (direct private rail express trains from Shinjuju station) and then straightforward to connect the next day to the shinkansen to Kyoto. There are a lot of lodging options, including ryokan, to chose from.

What the Hakone area offers is not just superb closeup views of Mt Fuji, but a variety of different types of transport which are all fun to ride -- old trains, cable cars, ships, ropeways. Check it out:

https://www.odakyu.jp/english/passes/hakone/

Note that since you will be only going one way from Tokyo, you would buy the pass at Odawara or Hakone Yumoto and get a regular ticket from Shinjuku to Hakone Yumoto.

It's a very different experience from Kyoto/Nara but one rhat is well worthwhile.