= Solution
At the fitted parameters, let $C=C_{\widehat\theta}$ and $\mu=\mu_{\widehat\theta}$. For prediction times $t_*$ define
$$
K_{**}=K_f(t_*,t_*),
\qquad
K_{*y}=\begin{pmatrix}K_f(t_*,t)&K_f(t_*,t-\widehat{\Delta t})\end{pmatrix}.
$$
The microlensing processes and measurement errors contribute no cross-covariance with the latent quasar light curve. The <Gaussian process regression posterior> is therefore
$$
\boxed{\mathbb E[f_*\mid y]=c\mathbf1+K_{*y}C^{-1}(y-\mu),}
$$
$$
\boxed{\operatorname{Cov}(f_*\mid y)
=K_{**}-K_{*y}C^{-1}K_{y*}.}
$$
The requested pointwise posterior variances are the diagonal entries of the latter matrix.
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