Consider Kikunoi - a 3-star Michelin restaurant, though we have no luck in securing a table for few times despite booking weeks in advance.
Otherwise, Kagaya serves a few Kaiseki courses during dinner time. It's on the dining floors of Isetan department store - on top of Kyoto station. We've been there once and it was nice.Edited: 09 September 2017, 17:39
Thanks, will check these out!
Jiki Miyazawa is not exactly Kaiseki restaurant, but Kappo restaurant serving particular Tofu with sesame flavored. It only has ten year history.
Kikunoi Main is the top notch traditional Kaiseki restaurant with 100/200 year history. Budget may be 30000 yen per person. If you are single customer, you will be rejected. It is a Ryotei restaurant with tea room separately situated: A Nakai server brings dishes of your course dinner one by one with the best temperature. If you are single customer, Kikunoi Roan, Kappo restaurant, may be accepted: location is behind Macdonald’s on Shijo and Kiyamachi street. Kappo is a kind of restaurant with counter seats and tables are situated. Budget is less than Ryotei Restaurant.
Kikunoi’s competitor, Kiccho also runs a Kappo restaurant in Gion, named Hana Kiccho. Both Kikunoi Main and Kiccho Arashiyama own Michelan three stars and five others own it. Booking is not easy: some of them require months or weeks in advance.
If your budget is 10000 yen for course dinner, it is not easy to locate traditional good Kaiseki. However, one or two is available. One is located at South Hanamikoji Street and the other couple of restaurants at Pontocho. However, I myself and Tripadviser folks never recommend such cheap Kaiseki.
How about Tempura? If you are hesitate to eat uncooked fishes, deep fried fishes may be easy to grab. Endoh is very popular tempura house even amongst tourists. Endoh runs three restaurants on Yasaka Street behind Kenniji Temple.