= Solution
Take $\Omega'$ nonempty and compactly contained in $\Omega$. The argument is a <compactness proof of a nonnegative elliptic solution estimate>, with the <global Schauder estimate> retaining control all the way to the boundary. The operator and the <Hölder space> exponent are fixed throughout.
Suppose the estimate fails. For each integer $k$ there is an admissible nonnegative solution $u_k$, with forcing $f_k$, such that
$$
M_k:=\sup_\Omega u_k>k\left(\inf_{\Omega'}u_k+\|f_k\|_{C^{0,\mu}(\overline\Omega)}\right).
$$
Put $v_k=u_k/M_k$ and $g_k=f_k/M_k$. Then $0\leq v_k\leq1$, $\max_{\overline\Omega}v_k=1$, $v_k=0$ on the boundary, $Lv_k=g_k$, and
$$
\inf_{\Omega'}v_k<\frac1k,\qquad
\|g_k\|_{C^{0,\mu}(\overline\Omega)}<\frac1k.
$$
Global <elliptic regularity> and the <global Schauder estimate> for zero <Dirichlet boundary condition> give
$$
\|v_k\|_{C^{2,\mu}(\overline\Omega)}\leq C\left(\|v_k\|_{C^0(\overline\Omega)}+\|g_k\|_{C^{0,\mu}(\overline\Omega)}\right)\leq2C.
$$
This estimate includes the $C^0$ term, so it does not require invertibility of $L$ or a sign restriction on $c$. The <compact embedding of Hölder spaces>, or the <Arzelà-Ascoli theorem> applied through second <partial derivatives>, gives a subsequence converging in $C^2(\overline\Omega)$ to $v$. Thus $v\geq0$, $Lv=0$, $v=0$ on the boundary, and $\max v=1$.
Choose $x_k\in\Omega'$ with $v_k(x_k)\leq\inf_{\Omega'}v_k+1/k$. A further subsequence has $x_k\to x_*\in\overline{\Omega'}\subset\Omega$. <Uniform convergence> yields $v(x_*)=0$. Part (b) says that a nonnegative homogeneous solution on the <connected> domain is either zero everywhere or strictly positive inside. Both alternatives contradict, respectively, $\max v=1$ and $v(x_*)=0$. Therefore
$$
\boxed{\sup_\Omega u\leq C(n,L,\Omega',\Omega)\left(\inf_{\Omega'}u+\|f\|_{C^{0,\mu}(\overline\Omega)}\right).}
$$
\b[The interior infimum and the forcing control the global supremum.] Global boundary control is crucial here: an interior-only <compactness> argument could lose the normalized maximum at the boundary.
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