I am interested in a simple markov chain with transition matrix $A$, parameterized by $p \in [0,1]$:
$$A = \begin{bmatrix} 0 & p & 1-p & 0 \\ 1-p & 0 & 0 & p \\ 0 & p & 1-p & 0 \\ 1-p & 0 & 0 & p \end{bmatrix}$$
I have a function $g$ on the state space $S = \{1,2,3,4\}:$ $$g(X_n) = \mathbb{P}(X_n = 1 \text{ or } X_n = 2).$$
The Markov Chain CLT says
$$\frac{S_n - n\mu}{\sigma\sqrt{n}} \to \mathcal{N}(0,1)$$
for some $\mu$ and $\sigma$ which I was able to calculate.
I want to use a Berry-Esseen type bound to get a number $n$ such that I need $n$ samples for this convergence in distribution to be "good enough". Anyone know any good papers or references for this? I have found a few (Mann 1996 - Berry-Esseen Central Limit Theorems For Markov Chains) and (Kloeckner 2018 - Effective limit theorems for Markov chains with a spectral gap) that have some good inequalities, but I had a lot of trouble using their inequalities to come up with explicit bounds.
Any help is appreciated.