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I sometimes do the reverse of this.
If you cut across at Oval, go through Kennington park, then along Cooks > Ruskin and then work your way across to Burgess Park, you can go through the park, cross OKR at the lights, and pick up the cycle routes along Willow Walk or Mandela Way and then go up onto BS .
Takes some fiddling to work out the optimal route but you can eliminate most major roads and go through two parks. If you go up Page's Walk you get to see Stompy the tank too, who has just had a fresh coat of paint.
I commute from Battersea to Bermondsey Street and I've just started using the Turret Grove -> Lansdowne Way bit of Cycle Network Route 3 as a way of avoiding Wandsworth Road, and I like it. I probably go down Larkhall a bit later than most (usually around 9.15), and I barely see another vehicle until the car park that is Lansdowne Way. There are some absolutely mental speed bumps on Turret Grove though.
I haven't found a way of completely eliminating CS7 around Oval, but I have successfully ditched Elephant & Castle now.