Here is a simple counterexample that is a bit too long for a comment. 

Consider two discrete-time Markov chains on $S=\{1,2,3\}$ with the following two transition matrices:
$$
P_1 = \begin{bmatrix} 0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0 \end{bmatrix} \;,
\quad 
P_2 = \begin{bmatrix} 0 & 1/2 & 1/2 \\
1/2 & 0 & 1/2 \\
1/2 & 1/2 & 0 \end{bmatrix}
$$
Both chains are irreducible and leave the uniform distribution invariant. The corresponding discrete-time Markov chains have the same lag-$1$ equilibrium autocorrelation, i.e.,  $\ell_1=-1/2$.  However, for $k>1$ their lag-$k$ autocorrelations  are quite different since the first chain is periodic with period $3$ while the second chain's lag-$k$ autocorrelation decays to zero with $k$.

This is not quite a counterexample, since the first chain does not have a steady-state or limiting  distribution.  To correct this deficiency, we break its periodicity by slightly perturbing its entries using a small parameter $\epsilon>0$:
$$
\tilde P_1 = \begin{bmatrix} 0 & 1-\epsilon & \epsilon \\
\epsilon & 0 & 1-\epsilon \\
1-\epsilon & \epsilon & 0 \end{bmatrix}
$$ The lag-$k$ correlation functions for the chains with transition matrices $\tilde P_1$ (blue line) and $P_2$ (black line) are plotted below with $\epsilon=1/25$ (chosen for visualization purposes only).  The inset shows the first few lag correlations.  Note that $\ell_1$ is almost the same for the two chains.  

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/TyFh3m.jpg