It seems Rosen's concept of "chasing" does invoke communication processes. Rosen doesn't speak much of Shannon at all in Life Itself; but he briefly cites him three times in another work, Essays on Life Itself. There, Rosen identifies Shannon's theory as a syntactical model. But I don't necessarily infer from that that Rosen disparages Shannon or his theory. For one thing, its very syntactical character is what lends the Shannon model its universal "portability" to all communicating systems alike in any language.
Boiling it all down, my sense (at this point!) is Rosen is working more on the semantic side of the problem, "What is life?" [in the sense that complex systems inevitably feature self-referential "loops" that cannot be reduced to purely syntactic terms because any instance of self-reference implies "meaning" [e.g., metabolism, repair, respiration, reproduction, etc., etc.), which is semantics, not syntax; and anyway, syntax is just "grammar"; thus it can eliminate all connection to "external referents" altogether, by which strategy it gains its universality — i.e., by sacrificing any idea of a "tie" between any particular system to any particular meaning]; Shannon simply focuses on one generic function: the successful communication of information. I.e., on the "medium" (syntax), not the "message" (semantics).
In any case, Shannon's syntactic model seems well suited to provide the rules that apply to the successful communication of a semantically richer model, be it Shakespeare's Hamlet or Robert Rosen's theory of relational biology — or any other communication whatever, including communications transpiring "inside" an organism itself.
I am pretty sure that if Rosen had a "problem" with anything Shannon did, he would not have hesitated to mention it. But so far, I have found no criticism. [But jeepers! You should see what he does to Jacques Monod! LOLOL!]
But then you never know — if, as I go through Essays, I should find a critical discussion of Shannon, I'll let you know!
Thank you for your magnificent essay/post, dearest sister in Christ! You lift everything up into the context of the spiritual — which ultimately seems to be the "largest model" (so to speak). [Rosen doesn't "go there"; but he seems very careful not to rule it out.]
But without it, there would be no model for the "chasing" in Rosen's model. Actually, Rosen implies as much by his appeals to Turing et al.