No, compatibilism only maintains that the everyday concept of what "free will" means is fully compatible with a deterministic chaotic reality. Human beings are basically "moist robots", to use a facetious metaphor. It does not maintain that free will is an illusion except when it is construed as freedom from causal determinism. That would be the antithesis of compatibilism.
I don't understand why you think that any branch of science, let alone scientific methodology, has anything of substance to say about "moral responsibility". That is a subject for philosophy, I think, and compatibilism implies that we are morally responsible for the consequences of our actions. Compatibilism itself has no essential connection with theism or theology, although some theists may subscribe to compatibilism. Dennett, a very well-known compatibilist, is a famous "new atheist"--one of the so-called
four horsemen of atheism.
I find it annoying that you continue to be unable to set off quoted material in order to distinguish is from your own contribution to the discussion. There are many ways to do that. You can use a quote box or just put the text in italics and indent it, as I have done in modifying the above quote from your post. Readers have a right to know that you are using quoted material rather than you own words, even if you agree with everything in that quoted material. Technically, it comes off as plagiarism, although you do cite the source. I just bothered to check how much of the material came from this blogger that you referenced.
Now, regarding that material that you quoted from that blog, the authors are advocates for libertarian free will, which is rejected by most compatibilists. Again, I think that this is just evidence that you really don't understand compatibilism and its difference from other competing positions in the philosophical debates over free will.
I think that you would consider the bolded material and its context relevant, if you actually had a good grasp of what chaos theory and compatibilism are about. I continue to believe that you don't know what you are talking about when you insert "chaos theory" into this discussion. I have no idea how you think it has anything to do with "free will" here except insofar as it is about chaotic deterministic systems. If free will is compatible with determinism, it is most certainly compatible with
chaotic determinism.