The unique measure of maximal entropy $\mu_f$ supported on the Julia set of a rational map $f$ of degree $d \geq 2$ is indeed the unique balanced measure for $f$, i.e., the only probability measure $\mu$ not charging the exceptional set and satisfying $f^*\mu =d \cdot \mu$. As you already noticed in the comments, uniqueness of a measure with this property is explicitly stated in the mentioned paper by Freire, Lopes and Ma\~ne in their Theorem, part (d) (page 46). The proof of this statement is on p. 55 and the argument goes as follows: for any balanced measure $\mu$ it is shown that $\mu$ is absolutely continuous with respect to $\mu_f$ and the ergodicity of $\mu_f$ implies that $\mu=\mu_f$ (existence and ergodicity of $\mu_f$ are proved earlier in the paper). No assumption of non-atomicity, no reference to critical points or classification of Fatou components is employed in the proof of this uniqueness statement.

Another way to prove uniqueness of balanced measure is to use potentials of measures on the Riemann sphere $\mathbb{C}_\infty$ introduced as in

F. Berteloot, V. Mayer, Rudiments de dynamique
holomorphe, Vol. 7 of Cours Sp\'ecialis\'es,
Soci\'et\'e Math\'ematique de France, Paris (2001)

They give a streamlined treatment based on prior results by Fornaess and Sibony, Hubbard and Papadopol, Ueda and others.
Consider the cone $\mathcal{P}$ of functions $U$ plurisubharmonic on $\mathbb{C}^2$ and satisfying $U(tz)=c\log|t|+U(z)$ with a constant $c=c(U) >0$. Each such function defines a positive measure $\mu_U$ on $\mathbb{C}_\infty$ by $\langle \mu_U, \Phi \rangle =\int_{\mathbb{C}_\infty}(U \circ \sigma)\frac{i}{\pi}\partial\bar{\partial}\Phi$ for every smooth test function $\Phi$ with support in the domain of definition of the section $\sigma$ of the natural projection $\Pi: \mathbb{C}^2\setminus \{0\} \to \mathbb{C}_\infty$. Furthermore, every positive measure $\nu$ on $\mathbb{C}_\infty$ is defined by a function $U \in \mathcal{P}$ (unique if required to satisfy $\sup_{\|z\|\leq 1}U(z)=0$), specifically by $U(z)=\int_{\mathbb{C}_\infty}\log\frac{|z_1w_2-z_2w_1|}{\|w\|}d\nu([w])$ (Th\'eor`eme VIII.9 in this reference). This is called the potential of $\nu$.

Now, if a measure $\nu$ is balanced, then its potential $U$ satisfies $F^*U=d\cdot U$ Lemme VIII.12), hence $\frac{1}{d^n}F^{*n}U=U$ for every $n$. Here $F$ denotes a lift of $F$ to $\mathbb{C}^2$. Taking limits in $L^1_{loc}$ as $n \to \infty$ we get $U=G_f$ (Th\'eor`eme VIII.15), the potential of the Lyubich-Freire-Lopes-Ma\~ne measure $\mu_f$. Lifts are not unique, but this does not cause a problem.

If you relax the assumption on a measure supported on Julia set to $f_*\mu = \mu$, then there can be more measures satisfying it, even ergodic ones, besides the measure $\mu_f$. Of course the entropy will be less than $\log d$, sometimes even $0$. For more details on this see

S. P. Lalley, Brownian motion and the equilibrium measure on the Julia
set of a rational mapping, Ann. Probab. 20, 4 (1992),
1932--1967.