Actually, Russ is right. It isn't just the snakes being there. It may be nigh impossible for the dog(s) to do the bouldering, scrambling, etc. that you do. WD is also right about the water level.
I think (but may be wrong) that at the appropriate flow, the river is runnable from the CA 4 bridge to the reservoir. Here's what I'd suggest, Google search for river runner groups doing the Moke(ulumne). Find a group that already knows the river and run it with them as a way to scout the river for potential backpacking later. I don't know if you can run decked C2s with camping gear stowed below deck on this water or not; I doubt it. If possible, a canoe camping trip might be a thought. I used to guide whitewater trips in C2 opens (high Class III bordering on IV) with floatation and C2 open camping trips on up to low Class III water. My back always hurt in a kayak and I don't know how well modern camping gear is stowable in modern kayaks.
Since WD mentioned Emigrant, specifically Kennedy Creek, you could easily do a loop going up Summit Creek to Emigrant Meadow, then back down Kennedy Creek. There are lakes and streams with fishing. I don't have any secret fishing places to offer, but on Summit Creek I know of a 30' x 8' swimming hole with cliff side walls and a 12 foot waterfall dropping straight into the pool, also a pool in granite that gets hot in the sun and basically provides a solar spa, and a secret secluded valley with a meandering stream through it that is one of the most beautiful places I have ever seen. I'm not going to post openly, but may share with those who PM me. Also in Emigrant, you can make all kinds of loops from Crabtree trailhead. In nearby Hoover, you could also go up the main fork of the West Walker River, climb to some lakes to the west, then hike cross country back down the West Fork of the West Walker. There are a number of lakes tangential to the loop. You also could go up the West Walker, climb to Dorothy Lake, go cross country to Tower Lake, then back down to the West Walker and out.
If you want details on any of these things, just ask; I have a bunch of maps already drawn on CalTopo.com.