Hi,

If you consider staying at the Tawaraya (I suppose in Kyoto) you must already know that it will cost you a lot. The Ryokan is very old and typical. The food is included (dinner and breakfast). The service is above the cloud.
I have found that going through a japanese travel agency like JTB will give you a slightly beter rate than booking direct. They (JTB) have buying power and can get you instant confirmation.
June is not a high season for Kyoto (rainy season in Japan) but any friday to saturday and saturday to sunday night will be more expensive AND difficult to book than a weekday night.
I highly recommend the experience but do not look for a cheap deal, all deals will be expensive.
Try to arrive mid afternoon and plan in leaving mid morning to enjoy the relaxing experience. It would be a waist to arrive late and leave early !
Enjoy