I'm not sure which campsites you are looking at. Camp 4 is no longer first come first serve, there's a lottery the day before. All the campgrounds in the valley are reservable and usually booked the first few minutes reservations are available. The backpacker camp in the valley is available even without a backpacker permit if you are arriving via public transportation - it's walk in, and you may end up sharing a bear locker and campsite but so far as I know you aren't going to be tossed out because it doesn't get "full" - they just keep packing people in. With a wilderness permit, you are given a night before and a night following your itinerary for backcountry travel.
Four night circuit around Yosemite as in the entire park, no. You can probably do four nights along the north rim of Yosemite Valley and drop down Snow Creek trail back to the valley floor, sure. That would give you some great overlooks of the Valley and all the granite.
https://www.backpacker.com/trips/alone- ... -north-rimYou can start from the valley floor at Rockslides trailhead, which is harder to find but still a permittable trailhead. Or catch a ride on a bus up the road toward Crane Flat and ask the driver to let you off at the Old Big Oak Flat trailhead parking on the way up.