This is more a comment than an answer. There are also details which use machinery I am uncomfortable with. I will list them later, maybe some expert can comment?

Let my try to answer the analogous question in the case of of etale cohomology $H^\bullet(X,\overline{ \mathbb Q_l})$, where $X_0$ is a variety over $\mathbb F_q$ and $X$ its basechange to the algebraic closure.
Using comparison results, one can maybe deduce the Hodge case, but I doubt that there is a strong enough result in literature. Or maybe one can proceed in a similar way using mixed Hodge modules, I am not sure either.

First of all we may compute $H^\bullet(X,\overline{ \mathbb Q_l})$ as the cohomology of the push forward $f_* \mathbb Q_l$ of the constant sheaf on $X$.

Now I it is true that the derived category $D^b(pt_0,\overline{ \mathbb Q_l})$ is just the derived category of finite dimensional $\overline{ \mathbb Q_l}$ vectorspaces equipped with a (quasi-unipotent) automorphism.
In other words it is (a block of) the category of finite dimensional $ \overline{ \mathbb Q_l}[x,x^{-1}]$ modules.

The latter category has cohomological dimension one, since $\overline{ \mathbb Q_l}[x,x^{-1}]$ is a principal ideal domain. In particular any complex is formal.

This means that
we find a quasi-isomorphism of complexes, $H^\bullet(X,\overline{ \mathbb Q_l}) \cong f_* \overline{ \mathbb Q_l}$
which is compatible with the Frobenius action!

Now transfering the $A_\infty$-structure along such a quasi-isomorphism, yields $m_n$ which are compatible with the Frobenius action. In particular they have the expected effect on weights i.e. just add them all up!

By the way observe that this is different to the effect of $m_n$ on cohomological degree (add up, add 2-n)! So if $H^\bullet(X,\overline{ \mathbb Q_l})$ is pure (e.g. $X$ smooth, proper) , it must be formal!

Ok, here are the fishy details I am not sure about:

First of all we need a dg-algebra structure on $f_* \overline{ \mathbb Q_l}$. For this one needs a dg-enhancement of the usual six functors or an infinity category version.

We want to transfer the dg-structure along $H^\bullet(X,\overline{ \mathbb Q_l}) \cong f_* \overline{ \mathbb Q_l}$. I only know how to transfer a dga structure along a homotopy retraction, I never learned how one does it with an arbitrary quasi-isomorphism or under what conditions it is possible.