Wonderful, and that sounds like a ideal time of year to visit, missing the over-whelming summer heat/humidity.
In cases such as yours I'd check Tripadvisor. I used to live in Tokyo and visited Kyoto but that was 20 years ago now. Things change fast there and even if I'd visited a couple of years ago I'd still start from scratch and cast a broad net to see what's on offer.
Picking a random week in October I came up with these search lists (I hope perhaps the URLs might bring up what I see for you):
Ignoring hotels for now, under 'Kyoto Specialty Lodging'
https://www.tripadvisor.com/Hotels-g298 ... otels.htmland under 'Kyoto Vacation Rentals'
https://www.tripadvisor.com/VacationRen ... ntals.htmlBut toggle the various options, read some reviews and see what reviewers say, note the location map and take it from there. Japan is never cheap but there are ways to seriously trim some of the costs, so some research beforehand will likely pay dividends.
I'd also have a good dig around on the JNTO-London site
https://www.seejapan.co.uk/