Notation of an LTI system consisting of LTI filters
As I've said in chat, I do not believe the system
$$
 \mathbf{y}(z) = \mathbf{C}(z) \mathbf{s}(z) + \mathbf{D}(z) \mathbf{x}(z) \\
 \mathbf{s}(z) = \mathbf{A}(z) \mathbf{s}(z) + \mathbf{B}(z) \mathbf{x}(z) ,
$$
with the matrices time-varying is an LTI system.

That means, I don't think we can answer your question.

If the matrices are not functions of $z$ then you can write the transfer function as:
$$
\mathbf{y}(z)/\mathbf{x}(z) = \mathbf{C}(zI - \mathbf{A})^{-1} \mathbf{B} + \mathbf{D}
$$
but it's not clear to me if this equation is valid if the matrices are time-varying / $z$-transforms.