Do as much as you can as early as you can
Consider napping in the afternoon and wandering some of the streets later in the evening. Gion, Higashiyama, Fushimi Inari, and downtown (e.g., Pontocho, Kiyamachi) are gorgeous at night, even if many of the attractions are closed.
Do shopping in the afternoons (e.g., Nishiki Market, Shinkyogoku, Teramachi, Sanjo, Demachimasugata, Furukawacho, Kyoto Station area, etc.)
Buy cooling wipes like these
Consider renting a bicycle (moving at speed keeps you a lot cooler than walking all the time
Consider indoor activities like the Railway Museum, Handicraft Center, and organized activities such as tea ceremony
Spend time down by the river (great on a bicycle) or up in places like Kurama/Kibune, Ohara, Mt. Hiei, etc. where the temps are cooler. Northern Higashiyama is also noticeably cooler than elsewhere in the city because of the flowing water (especially along the Philosopher's Path) and abundance of shade.
Not directly related, but go Gozan no Okuribi and Arashiyama Touro Nagashi on the 16th.

Finished a trip myself last week and the weather was already pretty shitty towards the end. Maybe not cooling off, but for staying comfortable:

Travel light: leave almost everything at your hotel or store it at a locker at a train station (200-500 yen/day).
Wear moisture-wicking synthetic clothes. I just wore gym T-shirts -- tight-fitting so I'd at least look half-presentable -- and Uniqlo Airism underwear, which certainly lives up to the hype. (Larger sizes are only sold online and at the Ginza flagship store, however.)
Plan your laundry. Probably won't be an issue for you, but I went to four cities in five days and I couldn't use the hotels' laundry services since they needed a full day to take care of it.
You can find cooling deodorant wipes at the convenience stores which are okay but I found them sort of pointless once it reached 30C+ and humid.

Related categories:

Reddit mentions of Gatsby Ice Deodorant Body Paper Ice Citrus 30 sheets