The answer to my own question
I just finished the Wonderland Trail around Rainier (great trip BTW). The only campground I found that would be problematic to hang a hammock was Mowich Lake. (I hiked it using a GoLite Eden I tent which worked great but I'd rather hammock it whenever possible.)
The Mowich Lake campground is a repurposed gravel parking lot. Ugly yes, but you do get a picnic table as a consolation prize. There are trees around the perimeter of some of the campground, so maybe the rangers would let you string up a hammock slightly out of bounds. Ask them first or you may piss them off.
I would recommend camping at Eagle's Roost instead of Mowich anyway. You can pass through Mowich (an extra 0.6 mile of walking) to grab your food cache and dump whatever trash you're carrying, take a dip in the lake, and relax a bit before continuing on to Eagle's Roost (if hiking clockwise). If going counter-clockwise, you can drop all your stuff in Eagle's Roost and do the short hike to Mowich and back just carrying your pack, trash, and some water shoes (the lake bottom has some pointy rocks).
But most of all...GO DO IT!