= Solution
Write $K=K_n$, $\epsilon=\epsilon_n$, and choose
$$
\delta=\frac{M\epsilon}{8}.
$$
Take a $\delta$-net $(\theta^1,\ldots,\theta^N)$ of $H_1$ in $\ell^2$, with centers in $H_1$. Since every alternative lies in the $K$-dimensional Euclidean ball of radius $n$, the <volumetric covering bound> gives
$$
N\leq\left(1+\frac{2n}{\delta}\right)^K,
\qquad
\log N\leq C K\log n
$$
for all large $n$. Put $v_j=\theta^j-\theta_0$ and define the <Gaussian net test>
$$
\Psi_n
=\mathbf1\left\{
\max_{j\leq N}\left(
\langle Y-\theta_0,v_j\rangle
-\frac12\|v_j\|_2^2
\right)\geq0
\right\}.
$$
The inner products are well-defined Gaussian linear functionals because $v_j\in\ell^2$. Under $H_0$,
$$
\langle Y-\theta_0,v_j\rangle
\sim N\left(0,\frac{\|v_j\|_2^2}{n}\right),
$$
and $\|v_j\|_2\geq M\epsilon$. A Gaussian tail bound and a union bound give
$$
\mathbb E_{\theta_0}\Psi_n
\leq N\exp\left(-\frac{nM^2\epsilon^2}{8}\right).
$$
For any $\theta\in H_1$, choose $j$ with $\|\theta-\theta^j\|_2\leq\delta$. Then
$$
\mathbb E_\theta\langle Y-\theta_0,v_j\rangle
=\langle\theta-\theta_0,v_j\rangle
\geq\|v_j\|_2^2-\delta\|v_j\|_2
\geq\frac78\|v_j\|_2^2.
$$
Another Gaussian tail bound gives
$$
\mathbb E_\theta(1-\Psi_n)
\leq\exp\left(-\frac{9n\|v_j\|_2^2}{128}\right)
\leq\exp\left(-\frac{9M^2K\log n}{128}\right).
$$
Because $n\epsilon_n^2=K_n\log n$, the entropy term in the type-I bound is dominated by the signal exponent when $M$ is sufficiently large. Also $\epsilon_n^2\to0$ implies $K_n<n$ eventually, so $\log n\geq\log K_n$. Given $c_1>0$, choose $M$ large enough to obtain
$$
\boxed{
\max\left\{
\mathbb E_{\theta_0}\Psi_n,
\sup_{\theta\in H_1}\mathbb E_\theta(1-\Psi_n)
\right\}
\leq e^{-c_1K_n\log K_n}.}
$$
Enlarging $M$ if necessary handles the finitely many initial $n$.
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