At long last, Karuizawa Prince Hotel East can be booked using Hyatt points.

https://www.hyatt.com/en-US/hotel/ja...tel-east/hndyk

Unfortunately, this hotel is rated a Cat 6, so not a very good value to be had with Hyatt points reservation. But if you're limiting yourself to using an international chain hotel, it's a better location than the Marriott option, and probably still cheaper than the Hilton option.