Nice quote Cue, I've never read a book by Dawkins, only seen his talks.
But Free, you don't need to distinguish between brain-consciousness and consciousness, or as I think you mean, between body and mind (Cartesian dualism). Here's the thing, if there is some supernatural feature of the mind, how could we possibly communicate that? The supernatural by definition cannot interact with the natural. Consciousness can't be supernatural, because we know it exists, and is responsive to cause and effect. Alternatively, you could suppose that certain laws of the universe are suspended within our minds, but all that would mean is that those 'laws' weren't entirely correct, and would be revised to make it so that whatever is going on in the brain is actually natural.
I think what you're looking for is a way to describe the specialness of consciousness, but we don't need magic to do that. We don't need to elevate consciousness to some higher plane in order for it to be special, for example, rainbows still look cool despite knowing that they're the result of light refracting off droplets. The complexity and mystery of its physical basis, the brain, is good enough. Even if consciousness were magical in some way, what would that change? It's not going to alter our desires, motivations, intentions, or any of that.