The first possibility is to replace weak convergence of the sequence $\mu_{n}$ with setwise convergence, that is $$ \int _ { A } f ( s ) \mu _ { n } ( d s ) \rightarrow \int _ { A } f ( s ) \mu ( d s ) \quad \text { as } \quad n \rightarrow + \infty, $$ for any bounded measurable function $f$, see p.231 of H. Royden, Real Analysis, 1968.
The second possibility is to keep the weak convergence of $\mu_{n}$ but to modify the right-hand side of the inequality as shown in
E. Feinberg, P. Kasyanov, N. Zadoianchuk, Fatou's lemma for weakly converging probabilities, Theory Probab. Appl. 58 (2014), no. 4, 683-689 (Arxiv1206.4073).
Theorem 1.1 p.2, stated for measurable functions instead of sets, says
Let $\{\mu_{n}\}\subset\mathbb{P}(A)$ converge weakly to $\mu\in\mathbb{P}(A)$, and let $\{f_{n}\}_{n\geq1}$ be a sequence of measurable nonnegative $\overline{\mathbb{R}}$-valued functions on $A$. Then $$ \int \liminf_{n\to\infty,~s'\to s}f _ { n } ( s ^ { \prime } ) \mu ( d s ) \leq \liminf _ { n \rightarrow \infty } \int f _ { n } ( s ) \mu _ { n } ( d s ). $$ or, equivalently, with $\limsup$ instead of $\liminf$, and a sequence $\{f_{n}\}$ of measurable functions uniformly bounded above, $$ \limsup_ { n \rightarrow \infty } \int f _ { n } ( s ) \mu _ { n } ( d s )\leq \int \limsup_{n\to\infty,~s'\to s}f _ { n } ( s ^ { \prime } ) \mu ( d s ). $$