= Solution
Define $\Delta t=t-t'$, $\mathbf r=\mathbf x-\mathbf x'$ in the three noncompact spatial directions, and $\Delta w=w-w'$. The compact source must be the periodic <Dirac delta>
$$
\delta_R(\Delta w)=\frac1R\sum_{n\in\mathbb Z}e^{ik_n\Delta w},\qquad k_n=\frac{2\pi n}R.
$$
For the wave operator as explicitly defined in the question, a <retarded circle-compactified massless propagator> is
$$
\boxed{G_{\mathrm{ret}}=\frac1R\sum_{n\in\mathbb Z}e^{ik_n\Delta w}
\int\frac{d^3k}{(2\pi)^3}\frac{d\omega}{2\pi}
\frac{e^{i\mathbf k\cdot\mathbf r-i\omega\Delta t}}
{\mathbf k^2+k_n^2-(\omega+i0)^2}.}
$$
The poles $\omega=\pm\sqrt{\mathbf k^2+k_n^2}-i0$ are both below the real axis, making the <integral> vanish for $\Delta t<0$. Performing just the frequency <integral> gives an equivalent expression
$$
G_{\mathrm{ret}}=\frac{\Theta(\Delta t)}R\sum_n e^{ik_n\Delta w}
\int\frac{d^3k}{(2\pi)^3}e^{i\mathbf k\cdot\mathbf r}
\frac{\sin(\Omega_n\Delta t)}{\Omega_n},
\qquad\Omega_n^2=\mathbf k^2+k_n^2.
$$
The first time <derivative> of $\Theta(t)\sin(\Omega t)/\Omega$ jumps by one at zero, so applying $\partial_t^2+\Omega^2$ yields $\delta(t)$. Together with the completeness relations for the <Fourier series> and <Fourier transform> this verifies the specified source and normalization. The symbol for the wave operator follows the question's explicit sign convention; it is the negative of $g^{AB}\nabla_A\nabla_B$ for its mostly-plus <metric tensor>.
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