Originally Posted by KI-NRT

Not only are those two places booked full, availability in most other Miyajima properties are slim to none as well. It's peak season and on a Saturday.

I did some digging, and there's availability at Miyajima Grand Hotel Arimoto. BUT... it's not bookable anywhere (Booking.com, Japanican, Rakuten Travel) except for JTB and Ikyu. There's a 99 square meter Japanese-Western room, as well as a 59 square meter sukiya-style room, both of which are far nicer than the standard rooms.
JTB and Ikyu are both Japanese booking sites and are only in Japanese, so you'll either need a nice hotel concierge (such as the one where you'll be staying in Tokyo) or have a friend book it.