= Solution
The required estimate follows from one-dimensional barriers on a sufficiently narrow slab. Put $\kappa=\sqrt\tau$. On $|x_1|<d_1<\pi/(2\kappa)$, the function
$$
h(x_1)=\frac{\cos(\kappa x_1)}{\cos(\kappa d_1)}
$$
satisfies $h''+\tau h=0$, equals one at $x_1=\pm d_1$, and obeys
$$
1\leq h\leq\frac1{\cos(\kappa d_1)}.
$$
Choose $\delta(\varepsilon,\tau)$ so that
$$
\cos(\sqrt\tau\,\delta)^{-1}\leq1+\varepsilon
$$
and so that the analogous solution of $k''+\tau k=-1$ with zero endpoint values is bounded by $e^{2d_1}-1$. Comparing $u$ and $-u$ with
$$
\left(\sup_{\partial\Omega_1}|u|\right)h
+\left(\sup_{\Omega_1}|f|\right)k,
\qquad
\Omega_1=\Omega\cap\{|x_1|<d_1\},
$$
and using $c\leq\tau$ gives
$$
\sup_{\Omega_1}|u|
\leq(1+\varepsilon)\sup_{\partial\Omega_1}|u|
+(e^{2d_1}-1)\sup_{\Omega_1}|f|.
$$
The permitted width is of order $\tau^{-1/2}$, so it can be chosen with $\delta(\varepsilon,\tau)\to\infty$ as $\tau\downarrow0$.
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