$\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$Take any $\mu\in\M(\Om)$, any open subset $\Th$ of $\Om$, and any real $\ep>0$. Let $\de:=\ep/4$.
By the Hahn decomposition theorem, there is a partition of $\Om$ into Borel sets $D^\pm$ such that $D^+$ is a positive set for $\mu$ and $D^-$ is a negative set for $\mu$.
Let
\begin{equation*}
A^\pm:=\Th\cap D^\pm. \tag{1}\label{1}
\end{equation*}
Since $|\mu|$ is inner regular, there exist compact sets
\begin{equation*}
K^\pm\subseteq A^\pm\text{ such that }|\mu|(A^\pm\setminus K^\pm)<\de. \tag{2}\label{2}
\end{equation*}
Since $\Om$ is normal, there exist open subsets $U^\pm$ of $\Th$ such that
\begin{equation*}
U^\pm\supseteq K^\pm\text{ and }U^+\cap U^-=\emptyset. \tag{3}\label{3}
\end{equation*}
Since the sets $K^\pm$ are compact and $\Om$ is locally compact, without loss of generality the closures of the sets $U^\pm$ are compact.
By Urysohn'slemma, there exist continuous functions $f^\pm\colon\Om\to\R$ such that
\begin{equation*}
0\le f^\pm\le1,\quad f^\pm=1\text{ on }K^\pm,\quad f^\pm=0\text{ on }\Om\setminus U^\pm. \tag{4}\label{4}
\end{equation*}
Let
\begin{equation*}
f:=f^+-f^-.
\end{equation*}
Then $f^+f^-=0$, whence $|f|\le1$. Also, $f=0$ on $\Om\setminus(U^+\cup U^-)$. So, recalling that the closures of the sets $U^\pm$ are compact, we see that $f\in C_c(\Om)$. Also, since $U^\pm$ are subsets of $\Th$, we have $|f|\le1_\Th$.
It remains to show that
\begin{equation*}
\int_\Om f\,d\mu\ge|\mu|(\Th)-\ep. \tag{$*$}\label{*}
\end{equation*}
To do this, note that, by \eqref{3}, \eqref{2}, and \eqref{1},
\begin{equation}
\begin{aligned}
|\mu|(U^-\setminus K^-)&\le|\mu|(\Th\setminus U^+\setminus K^-) \\
&=|\mu|(\Th)-|\mu|(U^+)-|\mu|(K^-) \\
&\le|\mu|(\Th)-|\mu|(K^+)-|\mu|(K^-) \\
&<|\mu|(\Th)-|\mu|(A^+)-|\mu|(A^-)+2\de=2\de.
\end{aligned}
\tag{5}\label{5}
\end{equation}
So, by \eqref{4}, \eqref{3}, \eqref{2}, \eqref{5}, and \eqref{1},
\begin{equation*}
\begin{aligned}
\int_\Om f\,d\mu&=\int_{U^+} f^+\,d\mu-\int_{U^-} f^-\,d\mu \\
&\ge\int_{K^+} f^+\,d\mu-\int_{K^-} f^-\,d\mu -\int_{U^-\setminus K^-} f^-\,d\mu \\
&=\mu(K^+)-\mu(K^-) -\int_{U^-\setminus K^-} f^-\,d\mu \\
&\ge\mu(K^+)-\mu(K^-) -|\mu|(U^-\setminus K^-) \\
&>\mu(A^+)-\de-\mu(A^-)-\de -2\de \\
&=|\mu|(\Th)-4\de=|\mu|(\Th)-\ep,
\end{aligned}
\end{equation*}
so that \eqref{*} is proved. $\quad\Box$
$\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$By the polar decomposition of complex measures, there is a Borel function $g\colon\Om\to[0,2\pi)$ such that
\begin{equation*}
|\mu|(\Th)=\int_\Om 1_\Th e^{ig}\,d\mu. \tag{1}\label{1}
\end{equation*}
Take any real $\ep>0$. Next, take any natural
\begin{equation*}
m>\frac{2\pi |\mu|(\Th)}{\ep/2} \tag{2}\label{2}
\end{equation*}
and any
\begin{equation*}
\de\in\Big(0,\frac\ep{2(2m+1)}\Big). \tag{3}\label{3}
\end{equation*}
For $j\in[m]:=\{1,\dots,m\}$, let
\begin{equation*}
I_j:=[\tfrac{2\pi(j-1)}m,\tfrac{2\pi j}m),\ A_j:=\Th\cap g^{-1}(I_j), \tag{4}\label{4}
\end{equation*}
so that the $A_j$'s are Borel sets forming a partition of $\Th$.
Since $\Om$ is a metric space and $|\mu|$ is a Borel measure, $|\mu|$ is regular. So, for each $j\in[m]$ there exist a closed set $F_j$ and an open set $G_j$ such that
\begin{equation*}
F_j\subseteq A_j\subseteq G_j\text{ and }|\mu|(G_j\setminus F_j)<\de, \tag{5}\label{5}
\end{equation*}
so that the $F_j$'s are (pairwise) disjoint and for
\begin{equation*}
F:=\bigcup_{j\in[m]}F_j\text{ and }G:=\Th\setminus F \tag{6}\label{6}
\end{equation*}
we have
\begin{equation*}
|\mu|(G)=\sum_{j\in[m]}|\mu|(A_j\setminus F_j)<m\de. \tag{8}\label{8}
\end{equation*}
All metric spaces are normal. So, by Urysohn's lemma, for each $j\in[m]$ there exists a continuous function $h_j\colon\Om\to\R$ such that
\begin{equation*}
h_j=1\text{ on }F_j,\ h_j=0\text{ on }G_j^c:=\Om\setminus G_j,\ 0\le h_j\le1. \tag{9}\label{9}
\end{equation*}
Let
\begin{equation*}
h:=\sum_{j\in[m]} \frac{2\pi j}m\,h_j. \tag{10}\label{10}
\end{equation*}
Then, by \eqref{6}, \eqref{10}, \eqref{9}, and \eqref{4}, on $F$ we have $0\le h-g\le\frac{2\pi}m$, and hence
\begin{equation*}
|e^{ih}-e^{ig}|\le\frac{2\pi}m\quad \text{on}\ F. \tag{11}\label{11}
\end{equation*}
Again by the regularity of $|\mu|$ and Urysohn's lemma, there exist a closed set $F_0$ and a continuous function $h_0\colon\Om\to\R$ such that
\begin{equation*}
F_0\subseteq\Th,\ |\mu|(\Th\setminus F_0)<\de, \tag{12}\label{12}
\end{equation*}
\begin{equation*}
h_0=1\text{ on }F_0,\ h_0=0\text{ on }\Th^c,\ 0\le h_0\le1. \tag{13}\label{13}
\end{equation*}
So, by \eqref{1}, \eqref{13}, \eqref{6}, \eqref{11}, \eqref{12}, \eqref{8}, \eqref{2}, \eqref{3},
\begin{equation*}
\begin{aligned}
&\Big||\mu|(\Th)-\int_\Om h_0 e^{ih}\,d\mu\Big| \\
=&\Big|\int_\Th e^{ig}\,d\mu-\int_\Th h_0 e^{ih}\,d\mu\Big| \\
\le&\int_\Th |1-h_0|\,d|\mu|+\int_\Th|e^{ig}-e^{ih}|\,d|\mu| \\
=&\int_\Th |1-h_0|\,d|\mu|+\int_F|e^{ig}-e^{ih}|\,d|\mu| +\int_G|e^{ig}-e^{ih}|\,d|\mu| \\
\le&|\mu|(\Th-F_0)+ \frac{2\pi}m\,|\mu|(F)+2|\mu|(G) \\
\le&\de+ \frac{2\pi}m\,|\mu|(\Th)+2m\de<\ep.
\end{aligned}
\end{equation*}
So,
\begin{equation*}
\begin{aligned}
|\mu|(\Th)&=\Re|\mu|(\Th) \\
&\le\ep+\Re\int_\Om h_0 e^{ih}\,d\mu \\
&=\ep+\lim_n\Re\int_\Om h_0 e^{ih}\,d\mu_n \\
&=\ep+\liminf_n\Re\int_\Om h_0 e^{ih}\,d\mu_n \\
&=\ep+\liminf_n\Re\int_\Th h_0 e^{ih}\,d\mu_n \\
&\le\ep+\liminf_n|\mu_n|(\Th).
\end{aligned}
\end{equation*}
Letting $\ep\downarrow0$, we conclude that
\begin{equation*}
|\mu|(\Th)\le\liminf_n|\mu_n|(\Th),
\end{equation*}
as desired.
Best Answer
A topology generated by countably many point-separating real functions is metrizable. To apply this here, it suffices to show that there is a countable family $\mathcal{G}$ of bounded real functions on $E$ such that the topology of weak convergence of measures is generated by the functions $\mu\mapsto g~\mathrm d\mu$ with $g$ in $\mathcal{G}$. That this is possible is Theorem 6.6 in "Probability measures on metric spaces" by Parthasarathy. The possibly first proof of this fact is in