So we know Claude is deterministic, but does that mean it is not conscious?

Or what is the reasoning exactly?

It largely comes down to how you define the term. Personally, I think anything that includes software (...of only tepid determinism, as we do explicitly add pseudorandomness) is not a particularly useful term.

Regardless, Dawkins seems to not have much interesting to add about the topic. A consistent theme for the last few decades, I must say.