I'm not a dualist, so bear that in mind, but I can see a couple of immediate logical objections to Carroll's argument.
First — and perhaps this is an unfair critique, even if accurate — Carroll is using a straw-man representation of religious beliefs. The idea of a 'personal' (identity preserving) soul that exists entirely separate from the physical plane is an unsophisticated conceptualization. Yes, it's a common conceptualization among both theists and atheists (which is why this critique may be unfair). But there are other ways of thinking about the biology/consciousness distinction than to treat consciousness as (a) an epiphenomenal byproduct or (b) a disconnected externality, but Carroll doesn't even hint at such. Why not think of consciousness as an emergent property of biology? Why not consider that the consciousness/biology relationship is a two-way street, not a one-way lane starting in biology? Either consideration would make his argument much harder, and as a result much more convincing if he succeeds.
That brings me to the second critique. Although I'll give him bonus point for invoking the Dirac equation, by using it Carroll is implying that the soul (or whatever) has to operate directly on electrons. This is that one-way lane approach: Carrol presumes that any 'spiritual' influence has to start deep down in the basic physics and biology of the brain and work its way up to influence consciousness. But it's well-known that the conscious mind can influence biology: people can slow their breathing or their pulse by act of will; a mere thought or perception can kick off the biological processes of anger, happiness, grief, etc. All we have to do is conceptualize that the 'spiritual' connects at the mental level and works its way down to influence the body, and the Dirac equation no longer applies.
Carroll's argument hinges on the preconception that consciousness is epiphenomenal and fully reducible to biology. That is not something that has been established by science, and without that preconception his argument doesn't have a lot of traction. I mean, I'm sure people will quibble with what I've said — I've just laid out a couple of critiques without building them into a full argument of their own — but I can't personally see a way around these critiques.