The camping options for the Bishop loop are many, from Forest Service campgrounds to RV parks and free/wild camping. One of my favorite spots is a Forest Service campground just outside Markleeville, one of the most beautiful places you'll ever see. The climb up Monitor Pass from 395 is stout and spectacular, one of the all-time greats, and the descent to Markleeville is a life-changing event. It's especially nice in the spring when snow is still on the high peaks.
Re. Sonoma start: I began in Santa Rosa, taking a bike path to Sebastapol and back roads to Occidental. From there, it's the spectacular and challenging Coleman Valley Rd., which takes you to the top of the coastal mountains and a big drop down to the coast a little north of Bodega Bay. Good campgrounds nearby.
To be honest, I haven't done the whole coast route in California. I've been from Crescent City all the way to Garberville, which includes the Avenue of the Giants, one of the great bike runs ever. I know the coast north of Fort Bragg, where I have cycled, tends too be narrow and hilly, but folks ride it all the time. Some non-cyclists freak out when I tell them I've ridden the Big Sur coast, but I've never had a problem. There is very little commercial traffic, and people interested in going fast will take Hwy 101 or Hwy 5 to the east. Great camping, too, all though I hated Plaskett Creek (sp?). Not a good hiker/biker.
If I had to pick a favorite, I would definitely choose the Bishop loop. The roads are more bike friendly, I like the big-ass mountains and passes, and the access to some free/wild camping makes for some good variety. I suggest, for best conditions, you ride it before Memorial Day or after Labor Day. Once the Bishop area is seeing temps in the 80's, you're set! Dang, I wish I were out there riding right now. Go do it.
A blog on that tour from 2009: https://www.crazyguyonabike.com/doc/page/?o=1&page_id=112686&v=2z