= Solution
We prove the <Interior second-derivative estimate for the Poisson equation> using <difference quotients>, rather than differentiating the $L^2$ forcing. Fix a coordinate $e_k$ and write $\delta_hv(x)=(v(x+he_k)-v(x))/h$. Choose a <smooth cutoff function> $\eta$ equal to one on $\Omega'$ with support compactly contained in $\Omega$, leaving room for all translations with sufficiently small $|h|$. We can arrange $|D\eta|\leq C(n)/d$, where $d=\operatorname{dist}(\Omega',\partial\Omega)$.
The standard difference quotient lemmas give discrete <integration by parts>, $\int a\delta_{-h}b=-\int(\delta_ha)b$, and $\|\delta_hv\|_2\leq\|D_kv\|_2$ on the relevant enlarged set. In the <weak formulation> $\int Du\cdot D\varphi=-\int f\varphi$, use the admissible <test function> $\varphi=-\delta_{-h}(\eta^2\delta_hu)$. Moving a difference quotient onto $Du$ gives
$$
\int\eta^2|D\delta_hu|^2+2\int\eta\delta_hu\,D\delta_hu\cdot D\eta=\int f\,\delta_{-h}(\eta^2\delta_hu).
$$
Set $X=\|\eta D\delta_hu\|_2$ and $Y=\|D\eta\,\delta_hu\|_2$. The cross term has absolute value at most $2XY$. Since $0\leq\eta\leq1$, the difference quotient lemma on the right gives
$$
\left|\int f\,\delta_{-h}(\eta^2\delta_hu)\right|\leq\|f\|_2\|D_k(\eta^2\delta_hu)\|_2\leq\|f\|_2(X+2Y).
$$
The <Young inequality> now implies $X^2\leq C(Y^2+\|f\|_2^2)$. Moreover $Y\leq C(n)d^{-1}\|Du\|_{L^2(\Omega)}$, independently of $h$. Hence, for every coordinate $k$,
$$
\|D\delta_hu\|_{L^2(\Omega')}\leq C(n,d)\left(\|Du\|_{L^2(\Omega)}+\|f\|_{L^2(\Omega)}\right).
$$
By the <Sobolev characterization by bounded difference quotients>, applied to each first <weak derivative> on smaller interior sets, these uniform bounds give all second <weak derivatives>. Their bounds pass to the weak limit as $h\to0$. Thus $u\in W^{2,2}_{\mathrm{loc}}(\Omega)$ and
$$
\boxed{\|u\|_{W^{2,2}(\Omega')}\leq C(n,d)\left(\|u\|_{W^{1,2}(\Omega)}+\|f\|_{L^2(\Omega)}\right).}
$$
No derivative of $f$ is used, and no boundary condition is needed for this interior <elliptic regularity> argument.
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