Glad you got your site at Fishing Bridge at this late date. I hate when folks state six months prior that "you're too late, the sites are sold by now". It always pays to check even if it's a few days prior. You never know.
We've used Mammoth quite often in June, August & Sept. We stayed nearby the previous night (you'll be right in the park) and get there by 9:00 or so. We've always gotten a site.
If a campground is marked as full at 7am then I would think all the sites have been paid for through that day by those on those sites already.
We have also gotten sites at Madison and Bridge Bay without reservations by being there early - 8 am - for cancellation or no-show sites.
Since you will be at Fishing Bridge you might just ask if you can just drive your truck to get a new site or how a campground can be full by 7am. However, since you have to leave Fishing Bridge anyway because your two night stay will be up, it really wouldn't make sense to drive to get another site and then come back for your RV - depending what time check-out is at Fishing Bridge. Then, too, you just might be able to pick up additional days at Fishing Bridge as long as you're there by cancellations, if that would work for you.
Good luck!
Full-Timed for 16 Years
.... Back in S&B Again
Traveled 8 yr in a 40' 2004 Newmar Dutch Star Motorhome
& 8 yr in a 33' Travel Supreme 5th Wheel