= Solution
Define $P_{\theta,n}=d^{-1}\sum_{S\in\mathcal S}P_{\theta,S}^{\otimes n}$. Expanding its <chi-squared divergence> by the <second moment of a mixture likelihood ratio>, and using the supplied one-observation cross moment and <independence>, gives
$$
\boxed{\chi^2(P_{\theta,n}\Vert P_0^{\otimes n})=\mathbb E_{S,T}\left[\left(1-\theta^2\frac{|S\cap T|^2}{k^2}\right)^{-n/2}\right]-1.}
$$
Here $S,T$ are independent uniform members of the specified family of cyclic intervals. Their overlap $R=|S\cap T|$ does not have the <hypergeometric distribution> of arbitrary uniform $k$-subsets. The family's geometry must be retained.
Let $x=\theta^2R^2/k^2<1/4$. Since $-\log(1-x)\leq x/(1-x)\leq2x$ and $R^2/k^2\leq R/k$, we obtain $(1-x)^{-n/2}\leq e^{nx}\leq e^{n\theta^2R/k}$. Taking the <expected value> proves the upper bound. The factor $n$ in the last exponent is present in the PDF and missing in the TeX aid.
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