It's hard to say. Bryce Canyon is very popular and you're suggesting June which could be when school kids are out and families traveling. We always found that if we were trying for FC/FS sites, early morning was going to best no matter what day. If we had to be somewhere popular at a specific time we'd make reservations and build the rest of our trip around that. You can always cancel. We were there in October and the weather was excellent even with snow one night and about 1/3-1/2 the CG was empty. But this was pre-2020 when things began to get crowded. My advice is if you really want to see Bryce Canyon then make reservations at RCC and cancel if plans change.

Which month will it be? I live in Utah and we always decide to camp last minute. I always arrive with several preplanned options (free and paid) in mind all preset in my CalTopo app for offline use. Sunday afternoon may be fine most months. There are back up options like the free dispersed FS/BLM lands and Coyote Hollow equestrian paid camp. Coyote Hollow was totally empty one October I visited, although other months it could be more filled with equestrian users. I'd so go for it, but have back ups already figured out just in case.

We were there late August, there were less than 10 sites occupied. There are also BLM sites between this CG and the turn off to Bryce NP.