Below is a maximum principle-based alternative to the proof of Mateusz.

We may assume that $r = 1$ by scaling. Let $w$ be the harmonic function on $B_1 \subset \mathbb{R}^n$ with boundary data $|\varphi|$. By the usual representation formula we have $$w|_{\partial B_{\frac{k+1}{2}}} \leq \frac{C(n,k)}{|\partial B_1|}\int_{\partial B_1}|\varphi| := A.$$ By the maximum principle, $|u| \leq w$ in $B_1 \backslash B_k$ (in particular, on $\partial B_{\frac{k+1}{2}}$). We conclude using the maximum principle again that $$|u| \leq A\frac{k^{2-n}-|x|^{2-n}}{k^{2-n}-\left(\frac{k+1}{2}\right)^{2-n}} := v$$ on $B_{\frac{k+1}{2}} \backslash B_k$. (Here we assumed $n \geq 3$; when $n = 2$, the function $v$ is obtained similarly using $\log$). Since $u = v = 0$ on $\partial B_k$ it follows that $$|u_{\nu}| \leq v_{\nu} = \frac{(n-2)k^{1-n}}{k^{2-n}-\left(\frac{k+1}{2}\right)^{2-n}}A$$ on $\partial B_k$, which is an estimate of the desired form.

Below is a maximum principle-based alternative to the proof of Mateusz.

We may assume that $r = 1$ by scaling. Let $w$ be the harmonic function on $B_1 \subset \mathbb{R}^n$ with boundary data $|\varphi|$. By the usual representation formula we have $$w|_{\partial B_{\frac{k+1}{2}}} \leq \frac{C(n,k)}{|\partial B_1|}\int_{\partial B_1}|\varphi| := A.$$ By the maximum principle, $|u| \leq w$ in $B_1 \backslash B_k$ (in particular, on $\partial B_{\frac{k+1}{2}}$). We conclude using the maximum principle again that $$|u| \leq A\frac{k^{2-n}-|x|^{2-n}}{k^{2-n}-\left(\frac{k+1}{2}\right)^{2-n}} := v$$ on $B_{\frac{k+1}{2}} \backslash B_k$. (Here we assumed $n \geq 3$; when $n = 2$, the function $v$ is obtained in similar way using $\log$). Since $u = v = 0$ on $\partial B_k$ it follows that $$|\nabla u| \leq |\nabla v| = \frac{(n-2)k^{1-n}}{k^{2-n}-\left(\frac{k+1}{2}\right)^{2-n}}A$$ on $\partial B_k$, which is an estimate of the desired form.