= Solution
Use a <bounded BV extension operator> for the bounded <Lipschitz domain>. Extend $u_k$ to $w_k\in BV(\mathbb R^n)$, supported in one fixed <compact set>, with $\|w_k\|_{BV(\mathbb R^n)}\le C_\Omega\|u_k\|_{BV(\Omega)}$. Such an extension is obtained by reflection in Lipschitz boundary charts, a <partition of unity> and a fixed <cutoff function>. A naive zero extension must also charge its boundary-trace jump; it cannot discard that contribution.
Let $M$ bound these <norms> and let $w_{k,\epsilon}=w_k*\rho_\epsilon$ be their <mollifications>. The <BV mollification error estimate> gives
$$
\|w_{k,\epsilon}-w_k\|_1\le\epsilon|Dw_k|(\mathbb R^n)\le\epsilon M.
$$
The <variation measure> here is on $\mathbb R^n$. The printed lemma's $|Dw|(\Omega)$ needs this correction unless it also assumes all the <derivative> mass lies in $\Omega$: a nonconstant bump supported outside $\overline\Omega$ disproves its literal wording. The correct estimate follows by averaging the <BV translation estimate> $\|w(\cdot-h)-w\|_1\le|h||Dw|(\mathbb R^n)$ over a mollifier supported in $|h|\le\epsilon$.
For each fixed $\epsilon>0$, <convolution> gives uniform bounds
$$
\|w_{k,\epsilon}\|_\infty\le M\|\rho_\epsilon\|_\infty,\qquad \|\nabla w_{k,\epsilon}\|_\infty\le M\|\nabla\rho_\epsilon\|_\infty.
$$
The <mollifications> have common <compact support> and are uniformly <equicontinuous>. The <Arzelà-Ascoli theorem> supplies a uniformly convergent <subsequence> at each scale $\epsilon=1/m$. A diagonal <subsequence> converges at every one of those scales. For two late members of that <subsequence>,
$$
\|w_k-w_\ell\|_1\le2M/m+\|w_{k,1/m}-w_{\ell,1/m}\|_1.
$$
First choose large $m$, then late $k,\ell$; the sequence is a <Cauchy sequence> in $L^1$. Its limit $w$ belongs to the <BV space> because the <total variation seminorm on a domain> is <sequentially lower semicontinuous>. Uniform <derivative> bounds and <integration by parts> give $Dw_k\overset{*}{\rightharpoonup}Dw$. Restriction to $\Omega$ proves the required <weak-star convergence in BV>.
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