I really genuinely think you have misunderstood the book, and Dawkins' philosophy in general.
Dawkins isn't a social darwinist (and nor was Darwin for that matter). He doesn't believe that nature red in tooth and claw is the right way for people to behave. He's argued in many places that, as self-aware thinkers, we have the capacity and moral duty to rise above it.
If you want a target for your arguments, you should pick on an actual social darwinist like Nietzsche.
Dawkins isn't a social darwinist (and nor was Darwin for that matter). He doesn't believe that nature red in tooth and claw is the right way for people to behave. He's argued in many places that, as self-aware thinkers, we have the capacity and moral duty to rise above it.
If you want a target for your arguments, you should pick on an actual social darwinist like Nietzsche.