I was in the same predicament at the Westin Miyako last week. I wanted to extend my multi-night stay by one night. After having no luck with Starwood, I asked the front desk at checkin and was told no availability whatsoever. A few hours later, I had a bit more luck at the Miyako satellite check-in in the Kyoto train station. The agent made a phone call to the "reservations office" at the Miyako, and after a few minutes conversation in Japanese (we couldn't understand) came back and offered us two rooms each with a twin bed. We declined, but we were pleased that he had tried a bit harder than the front desk.

I'm not sure if the "reservations office" is just the front desk, or if it truly is a separate office, but you may have better luck either by inquiring at the satellite office or by insisting to speak with their reservations office.

As another side note, if you're a gold, preferred plus, or corporate preferred member, you get free 4pm checkout. I was able to take advantage of this to extend my stay a bit with no problems at the Miyako even though the hotel was humming with activity on their "sold-out" night. And regardless of your status, you could have your bags sent to the satellite check-in at the Kyoto station where they can be held until they close at 6pm. This might make it a bit easier for you to take advantage of your last day sightseeing in Kyoto.

Good luck!