I've had a look at most major and minor booking websites and it is indeed hard to find anything at all, something I had not expected. I Have truly no idea what could be the cause, expect for the Sakura.
Since Kyoto is off the table, I'd advise looking at alternatives. Especially places that are on the Shinkansen - since you can use the JR Pass to travel on them at any time. So I did a couple of searches on nearby hotels for the period specified for 3 persons / 7 days and it was pretty hard to find anything to match to these criteria.
My advise would to try and look for a shorter period of time in a hotel 3 or 4 days or maybe even less and change hotels once or twice, this makes it easier to find availability.
Do you want one room for the 3 of you or separate rooms?