I'm a little confused. You're posting in September that you have a permit in August. Since I know they don't give permits more than 5 months in advance, I know it's not August 2022 that you're talking about. Where did you get your time machine? I want one! (You might mean you got one in August, I realize, but ...)
You're going to be hiking uphill for a good 2/3 of the trip, but at least it's gradual. The campsites in the canyon will have changed a lot - falling trees, fire, etc rewrite the landscape. There are fewer spots than years past. But, it's late in the year and you'll likely have little competition. I wouldn't worry about it overmuch. If I were doing it that direction I would camp at the top of the drop into Pate - good sites at Morrison Creek with great views - at Register Creek, and at Waterwheel or Glen Aulin. Once you're past Glen Aulin there's no legal camping.
The main thing the rangers care about is that you leave on the date you're supposed to leave. There's no quota on the exit date. When you pick up the permit, you can mention that you're not sure about getting out on time so want to have an extra day, that will be no issue at all.