This looks somewhat similar to what I was thinking and the attempt at formalization seems helpful. But it's hard for me to be sure. It's hard for me to understand the conceptual meaning and implications of it. What are your own thoughts on your formalization there?

I'm inclined to think that it doesn't really show anything metaphysically significant. When we encode facts about S as propositions, we are conceptually slicing and dicing the-way-S-is into discrete features for our map of S. No matter how we had sliced up the-way-S-is, we would have gotten a collection of features encoded as proposition. Finer or coarser slicings would have given us more or less specific propositions (i.e., propositions that pick out minuter details).

When we put those propositions back together with propositional formulas, we are, in some sense, recombining some of the features to describe a finer or coarser fact about the system. The fact that T is closed under all the formulas in C just says that, when we slice up the-way-S-is, and then recombine some of the slices, what we get is just another slice of the-way-S-is. In other words, my remark about T and C is just part of what it means to pick out particular features of a physical system.