Preliminaries and Standard Technical Framework
Let $T \in (0, \infty)$ be fixed.
Let $d \in \mathbb{N}_{\geq 1}$ be fixed.
Let $$(\Omega, \mathcal{G}, (\mathcal{G}_t)_{t \in [0,T]}, \mathbb{P})$$ be a complete probability space with a complete, right-continuous filtration $(\mathcal{G}_t)_{t \in [0,T]}$.
Let $$B : [0,T] \times \Omega \rightarrow \mathbb{R}^d, \quad (t,\omega) \mapsto B_t(\omega)$$ be a standard $d$-dimensional $(\mathcal{G}_t)_{t \in [0,T]}$-adapted Brownian motion on $\mathbb{R}^d$ such that, for every pair $(t,s) \in \mathbb{R}^2$ with $0 \leq t < s$, the random variable $B_s-B_t$ is independent of $\mathcal{G}_t$.
Let \begin{align} &\sigma: \mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}, \\ &\mu: \mathbb{R}^d \rightarrow \mathbb{R}^{d}, \end{align} be affine linear transformations, i.e. let there be matrices $(A^{(\sigma)}_1,...,A^{(\sigma)}_d, \bar{A}^{(\sigma)}):= \theta_{\sigma} \in (\mathbb{R}^{d \times d})^{d+1}$ such that, for all $x \in \mathbb{R}^d$, \begin{equation} \sigma(x) = ( A^{(\sigma)}_1 x \mid ... \mid A^{(\sigma)}_d x) + \bar{A}^{(\sigma)}, \end{equation} where $A^{(\sigma)}_i x$ describes the $i$-th column of the matrix $\sigma(x) \in \mathbb{R}^{d \times d}$, and let there be a matrix-vector pair $(A^{(\mu)}, \bar{a}^{(\mu)}) := \theta_{\mu} \in \mathbb{R}^{d \times d} \times \mathbb{R}^d$ such that, for all $x \in \mathbb{R}^d$, \begin{equation} \mu (x) = A^{(\mu)}x + \bar{a}^{(\mu)}. \end{equation} Note, that this implies, that $\sigma$ and $\mu$ are globally Lipschitz.
Let $x_0 \in \mathbb{R}^d$ be fixed.
Question
Consider the following stochastic differential equation, given as an equivalent stochastic integral equation, where the multidimensional integrals are to be read componentwise:
\begin{equation} S_t = x_0 + \int_{0}^{t} \mu(S_t) ds + \int_{0}^{t} \sigma (S_t) dB_s. \end{equation}
Under our assumptions, it is the case that an (up to indistinguishability) unique solution process
$$ S^{(x_0, \theta_{\sigma}, \theta_{\mu})} :[0,T] \times \Omega \rightarrow \mathbb{R}^d, \quad (t, \omega) \mapsto S_t(\omega),$$
for this equation exists (to see this, consider for example Theorem 8.3. in Brownian Motion, Martingales and Stochastic Calculus from Le Gall).
Now, how can I construct a measurable mapping
$$(x_0, \theta_{\sigma}, \theta_{\mu}, t, \omega) \mapsto S^{(x_0, \theta_{\sigma}, \theta_{\mu})}_t(\omega) \in \mathbb{R}^d \quad ? $$
In particular, I have problems dealing with the fact, that $S^{(x_0, \theta_{\sigma}, \theta_{\mu})}$ is only unique up to indistinguishability. How cam I deal with this and "chose" a concrete process $S^{(x_0, \theta_{\sigma}, \theta_{\mu})}$ for each triple $(x_0, \theta_{\sigma}, \theta_{\mu})$ before evaluating this process at $(t, \omega)$?
This question is at the core of my current research. I am stuck and I would be extremely grateful for any advice!