= Solution
Define $\alpha=\hbar T/(2I)$. The free <propagator> acts diagonally on each angular harmonic:
$$
U_Te^{im\theta}=e^{-i\alpha m^2}e^{im\theta}.
$$
The kick multiplies by the Jacobi-Anger series
$$
e^{i\eta\cos\theta}=\sum_{\ell\in\mathbb Z}i^\ell J_\ell(\eta)e^{i\ell\theta}.
$$
Multiply the two series and collect the output harmonic $n=m+\ell$. The one-step coefficient update is
$$
\boxed{c_n(t+T)=\sum_{m\in\mathbb Z}
i^{\,n-m}J_{n-m}(\eta)e^{-i\alpha m^2}c_m(t),}
$$
where both sampling times are just after their respective kicks. Free evolution supplies the input-index <phase> before the kick mixes angular momenta.
The source uses the unnormalized basis functions $e^{im\theta}$. Thus a unit-norm <wavefunction> satisfies $2\pi\sum_m|c_m|^2=1$. If the basis is instead $e^{im\theta}/\sqrt{2\pi}$, its amplitudes have sum of squared moduli one and obey the same <matrix> update. Mixing these normalization conventions would change an energy expectation incorrectly.
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