If you launch on Brown's after about 3 pm you'll have the river to yourself. Better yet, launch at Wilderness Aware ($3 fee) and start with the Milk Run. Float down and find a campsite below Ruby Mt. Campground on public land. Continue to Stone Bridge the next am. If you get on the river before about 8:30 or so, you may be able to stay ahead of the train of rubber buses...
Good luck,
--Andy