= Solution
A fixed smooth choice $g\equiv0$ already gives a counterexample. Select nonzero smooth functions $\phi(x_1,x_2)$ and $\chi(x_3)$ with product support compactly contained in a small box inside the shell. Let
$$
u_k(x)=\phi(x_1,x_2)\chi(x_3)\sin(kx_3),\qquad
f_k=(\partial_1^2+\partial_2^2)\phi\,\chi\sin(kx_3).
$$
Then $u_k$ has zero boundary trace and is a classical, hence weak, solution. The $L^2$ <norms> of $f_k$ are bounded independently of $k$. On the other hand,
$$
\partial_3u_k=\phi\bigl[\chi'\sin(kx_3)+k\chi\cos(kx_3)\bigr].
$$
The squared <norm> of $\chi\cos(kx_3)$ tends to $\frac12\|\chi\|_2^2$, by the oscillatory integral identity for $\cos^2$. The other term has bounded <norm>, so $\|u_k\|_{H^1}$ grows at least proportionally to $k$. Therefore
$$
\boxed{\text{no uniform }C\text{ can satisfy the claimed estimate for this fixed }g=0.}
$$
This <failure of a coercive Dirichlet estimate under degeneracy> loses control in the third direction; it does not rely on varying $g$ with the sequence.
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