I think the odds of fog or snow on the North Rim are pretty low. Not impossible mind you but fall is normally a dry season on the Colorado Plateau.
Worst case scenario if Antelope canyon is on the menu is driving a few hours south from there and sliding in the east entrance.
The Capital Reef idea is good too.
There's a good lesson in that Ruby's story, it is worth calling ahead.
Accommodations of any type can be in short supply.
At Yosemite, Bryce, Death Valley, and the Grand Canyon accomodations can on any given day be completely full.