= Solution
The posterior is
$$
p(\mu,\theta\mid y,t)\propto
p(y\mid t;\mu,\theta)p(\mu,\theta).
$$
A random-walk <Metropolis–Hastings algorithm> proposes $\vartheta'$ from a symmetric density about the current $\vartheta=(\mu,\theta)$ and accepts with probability
$$
a(\vartheta,\vartheta')
=\min\left\{1,\frac{p(\vartheta'\mid y,t)}{p(\vartheta\mid y,t)}\right\}.
$$
For distinct states, multiplying the transition density by the target density gives
$$
p(\vartheta\mid y,t)q(\vartheta'\mid\vartheta)a(\vartheta,\vartheta')
=\min\{p(\vartheta\mid y,t)q,\ p(\vartheta'\mid y,t)q\},
$$
which is symmetric and proves <detailed balance>. Run multiple dispersed chains, tune proposals during warm-up, inspect traces, effective sample sizes and convergence diagnostics, then estimate the period mean by averaging $2\pi/\omega_m$ over retained draws.
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