= Solution
The angular <action> is $S=\int I\dot\theta^2ds/2$. In each winding sector, write $\theta=\theta_p+\xi$ with $\xi$ zero at both endpoints. The cross term vanishes and the Gaussian fluctuation <integral> is the free-particle prefactor with <mass> replaced by $I$. Thus
$$
\boxed{K_\theta(\Delta,T)=\sqrt{\frac{I}{2\pi i\hbar T}}\sum_{p\in\mathbb Z}\exp\left[\frac{iI(\Delta+2\pi p)^2}{2\hbar T}\right]}.
$$
Equivalently, multiply the line kernel for displacement $R(\Delta+2\pi p)$ by $R$ to change its measure from arc length to angle. The quadratic <action> makes the classical-path sum exact, rather than merely a <stationary phase> approximation. Real-time sums are understood with the usual negative imaginary part of $T$ and its boundary limit.
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