We just ran the run this past weekend. There were plenty of campsites along the way. We camped at a nice beach site just across from the cliffs at the warm spring. Easy swim across to the springs, or you could just ferry it on your raft. The train was definitely loud, as were some of the late night campers that hiked in from above and were camped on top of the cliffs, and we would probably camp elsewhere next time, but it was a fun camp right near the springs. The springs pool was just a little warmer than the river, but will probably warm up more as the river drops.
Further down, Cabin camps and the Island camps looked really nice, but we were only out for one night so didn't stay. Also saw multiple other unnamed camps on other islands and on the banks too, that looked great with shade.
I would not recommend setting up a tent at a spot unless you are staying there that same night. In my opinion, leaving your stuff at camp all set up is fine if you are running laps.