When I first read this question I was quite dismissive. However, I re-read The Emperor's New Mind by Penrose in the last week. Now I'm pausing to consider.
The basic idea of Bell's theorem is that, for certain specific types of measurement, it is possible to distinguish between quantum mechanics and a very broad range of deterministic theories called "hidden variable" theories.
This type of theory says that reality is really deterministic. It's just that there are control factors that we cannot see. These factors, if we could see them, would determine the outcome of experiments that, in quantum mechanics, are said to be randomly selected from a distribution. The control factors are called hidden variables.
Bell's theorem allows us to perform experiments that distinguish between quantum mechanics and hidden variables. There are situations that are physically possible, in which any hidden variable theory will predict results that are different from what QM will predict. And, in several well known experiments, QM has been shown to be the "victor."
Penrose's thesis is that consciousness can not be algorithmic. That is, he claims that digital computers, being entirely deterministic, will not be able to produce the equivalent of human thought. He then speculates (in a manner I found interesting) about how it could arise in matter such as brains. One of the speculations he proposes is that quantum mechanics is required for human consciousness.
Some of the issues Penrose touches are things such as Goedel's theorem and self reference, the Turing test, existing tech w.r.t. AI, etc.
One section of the book deals with measurements of nerve impulses in various situations, including direct stimulus of the brain of volunteer subjects. There are some unusual results about the time of build up of a nerve signal (indicating the subject is mentally preparing for an action) as compared to the time the subject reports preparing. And these are then compared with the subject's response to surprise stimulus, with the result that there is very much less prep time. This seems to indicate that preparation takes longer than we are aware of, but is not always required. Thus, quite ironically, indicating that a substantial part of consciousness is unconscious.
There is a great deal more than I am explaining, but this post is already growing.
As to free will and a "Bell's test": It would require a specific theory of what constituted free will, and how it connected to matter. That is, it would not be a context free test with Result A meaning free will, Result not-A meaning no free will. We could only test a specific hypothesis.
For example: Suppose we speculate, along with Penrose, that free will arises due to quantum mechanics. (I still have the problem that I have not given any useful definition of free will.) In principle we could test this speculation by attempting to construct a device explicitly based on (somewhat) hidden variables. That is, a completely deterministic device such as a digital computer. If we could p+roduce a digitial computer device that could exhibit free will (to whatever definition we have yet to supply) then it would put a lot of pressure on the speculation that free will requires QM.