= Solution
Let $\phi_d(z)=(2\pi)^{-d/2}\exp(-\|z\|^2/2)$ be the standard <multivariate normal density>. Independence gives the <proposal distribution> density
$$
q(x,y)=\int\phi_d(y-Ax)\,\nu(dA).
$$
For an <orthogonal matrix> $A$,
$$
\|y-Ax\|=\|A^Ty-x\|=\|x-A^Ty\|.
$$
The isotropic Gaussian density is unchanged by this transformation. Since $A^T$ has the same distribution as $A$,
$$
q(x,y)=\int\phi_d(x-A^Ty)\,\nu(dA)=\int\phi_d(x-Ay)\,\nu(dA)=q(y,x).
$$
Thus this is an <orthogonal-mixture Metropolis proposal> with a symmetric density; inversion invariance, rather than any assumption of a uniform distribution on orthogonal matrices, is what is needed.
With $\alpha(x,y)=\min\{1,\pi(y)/\pi(x)\}$, the accepted off-diagonal transitions satisfy
$$
\pi(x)q(x,y)\alpha(x,y)=q(x,y)\min\{\pi(x),\pi(y)\}
=\pi(y)q(y,x)\alpha(y,x).
$$
The rejection part is supported on the diagonal and satisfies <detailed balance> automatically. Values of the acceptance rule at starting points where $\pi(x)=0$ can be specified separately; they do not affect stationarity under $\pi$. Therefore the <Metropolis–Hastings algorithm> kernel is reversible with invariant density $\pi$, proving
$$
\boxed{\pi\text{ is a stationary distribution}.}
$$
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