I will try to provide an answer for a particular case of your last question: Let us consider (following my comment above) the case of $H=\mathbb{CZ}_2$ i.e. the group hopf algebra equipped with its non-trivial quasitriangular structure, that is the $R$-matrix $R=\frac{1}{2}(1\otimes 1+1\otimes g+g\otimes 1-g\otimes g)$.
Its category of representations $\mathbb{CZ}_2\text{-Mod}$, consists of the $\mathbb{Z}_2$-graded vector spaces, commonly refered to as super-vector spaces, with morphisms the even linear maps and an algebra $A$ in the category $\mathbb{CZ}_2\text{-Mod}$ is a $\mathbb{Z}_2$-graded associative algebra (or: associative superalgebra). The braiding of the category is provided by the $R$-matrix.
Consider a f.d. super-module ${}_{A}M$ and the super version of the interwiner, that is an homogeneous linear map $g\in \mathcal{E}nd_\beta(M)$,of degree $\beta(=0,1)$ which super-commutes with the $A$-action on $M$, that is:
$$
g(a\cdot m)=(-1)^{\beta\gamma}a\cdot g(m)
$$
for all $a\in A_\gamma$, $\gamma=0,1$, $m\in M$. Notice that if $\beta=0$ then $g$ is a usual super module morphism (that is an even super module homomorphism) while if $\beta=1$ then we get $\mathbb{C}$-linear, odd maps which are antilinear in the $A$-action. The set of all super-interwiners forms a superalgebra $$\mathcal{E}nd(M)=\mathcal{E}nd_0(M)\oplus\mathcal{E}nd_1(M)$$
In case ${}_{A}M$ is simple (in the super sense, i.e. it contains no proper $\mathbb{Z}_2$-graded submodules), we also have a super-version ($\mathbb{Z}_2$-graded version) of Schur's lemma:
$\mathcal{E}nd_0(M)=\mathbb{C}\cdot Id$ and $\mathcal{E}nd_1(M)=\mathbb{C}\cdot\theta$, where $\theta=0$ or $\theta$ is an odd linear map with $\theta^2=Id$
Considering the usual definitions for the super-center $Z_s(A)$, the super-commutative superalgebra $A$ and the super version of Schur's lemma, we get that:
If $A$ is a superalgebra, ${}_{A}M$ is a f.d. simple super-module and $z_\xi$ an homogeneous element in the supercenter $Z_s(A)$ then there exists some $\lambda=\lambda(z_\xi)\in\mathbb{C}$ such that either $z_\xi\cdot m=\lambda m$ (if $\xi=0$ i.e. $z$ is even) or $z_\xi\cdot m=\lambda\theta(m)$ (if $\xi=1$ i.e. if $z$ is odd), for all $m\in M$.
Finally, using the last lemma, we can get that:
Any f.d., simple super-module ${}_{A}M$ over the super-commutative superalgebra $A$ is at most $2$-dimensional.
P.S.: If you want similar results for more general braided monoidal categories (let say over the category of modules of some more general quasitriangular hopf algebra -even for some other group hopf algebra), i think we will need a braided generalization of Schur's lemma.
I am not aware if something like that exists in the literature (and i would be very interested to know if somebody else knows of such results).
