= Solution
Let $I=mR^2$, $T=t-t'>0$, and $\Delta=\theta-\theta'$. Periodic angular <wave functions> have modes $e^{il\theta}/\sqrt{2\pi}$ with integer $l$ and <energies> $\hbar^2l^2/(2I)$. Normalizing the kernel with respect to $d\theta'$ gives
$$
\boxed{K_\theta(\Delta,T)=\frac1{2\pi}\sum_{l\in\mathbb Z}e^{il\Delta-i\hbar l^2T/(2I)}}.
$$
If the <integration> measure is arc length $R\,d\theta'$, the corresponding kernel is $K_\theta/R$. This measure choice accounts for the otherwise ambiguous <radius> prefactor.
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