This answer is an expansion on the answer of Yuval Peres which addresses the concerns raised in my comment.
We will show the existence of a progressively measurable stochastic process $t\mapsto x^*(t)\in\mathbb{T}^2$ satisfying
$$|\Phi(t,x^*(t,\omega),\omega)-c(t,\omega)|^2 - R(t,\omega) =0 \tag{1}$$
in several steps.
Let $\mathcal{K}(\mathbb{T}^2)$ denote the set of non-empty compact subsets of $\mathbb{T}^2$ equipped with the Hausdorff metric $d_H$. It is well-known that $(\mathcal{K}(\mathbb{T}^2), d_H)$ is a compact metric space. We turn $\mathcal{K}(\mathbb{T}^2)$ into a measurable space by endowing it with the Borel $\sigma$-algebra $\mathcal{B}(\mathcal{K}(\mathbb{T}^2))$.
Lemma 1. The $\mathcal{K}(\mathbb{T}^2)$-valued stochastic process $t \mapsto \bar{B}(c(t), R(t))$ is progressively measurable.
Proof. Since $c(\cdot)$ and $R(\cdot)$ are both $\mathbb{T}^2$- and $\mathbb{R}$-valued progressively measurable processes respectively, it follows that that $(c(\cdot),R(\cdot))$ is also progressively measurable. It follows from the triangle inequality that the map
$$(\mathbb{T}^2, [0,\infty))\rightarrow (\mathcal{K}(\mathbb{T}^2),d_H), \quad (c,R)\mapsto \partial B(c,R)$$
is continuous, which in turn implies the desired conclusion. $\Box$
Although I omitted this detail in my original post, in application the stochastic flow $\Phi$ has the following property: for almost every $\omega$, the map $x\mapsto \Phi(t,x,\omega)$ is a homeomorphism of $\mathbb{T}^2$ for all $t\geq 0$. Moreover, the $C(\mathbb{T}^2)$-valued stochastic process
$$t \mapsto \Phi^{-1}(t,\cdot)\in C(\mathbb{T}^2)$$
is progressively measurable. This fact leads us to the next lemma, the proof of which is an easy exercise.
Lemma 2. The map
$$\Psi_1: (C(\mathbb{T}^2),\|\cdot\|_\infty) \times (\mathcal{K}(\mathbb{T}^2),d_H) \rightarrow (\mathcal{K}(\mathbb{T}^2),d_H), \qquad (f,A)\mapsto f(A)$$
is continuous.
We claim that the $\mathcal{K}(\mathbb{T}^2)$-valued process
$$\tilde{K}(t) := \Phi^{-1}(t,\cdot)(\partial B(c(t),R(t))):= \{y\in\mathbb{T}^2: y=\Phi^{-1}(t,x) \text{ for some } x\in \partial B(c(t),R(t))\}$$
is progressively measurable. Indeed, fix $t\geq 0$. Evidently, the map
$$\Psi_2:[0,t]\times \Omega \rightarrow C(\mathbb{T}^2)\times \mathcal{K}(\mathbb{T}^2), \quad (s,\omega) \mapsto (\Phi^{-1}(s,\cdot,\omega),\partial B(c(s,\omega), R(s,\omega)))$$
is $\mathcal{B}([0,t]) \otimes \mathcal{B}(\mathcal{K}(\mathbb{T}^2))$-measurable. So the claim follows from writing
$$\tilde{K}(s,\omega) = \Psi_1\circ \Psi_2(s,\omega)$$
and using Lemma 2.
Now $\tilde{K}(t,\omega)$ isn't quite the set $K(t,\omega)$ from Yuval Peres's answer (it's larger); however, this doesn't matter for the following reason. If $x\in\tilde{K}(t,\omega)$, then I claim that $|x|\geq r$. Otherwise, since $\Phi(t,\cdot,\omega)$ is a homeomorphism, we could find an open ball $B(x,\delta)\subset B(0,r)$ such that $\Phi(t,\cdot,\omega)(B(x,\delta))$ is open. But by definition of $\tilde{K}(t,\omega)$, we have that $|\Phi(t,x,\omega)-c(t,\omega)|=R(t,\omega)$, which implies that there exists $y\in B(x,\delta)$ such that
$$|\Phi(t,y,\omega)-c(t,\omega)|>R(t,\omega).$$
This last inequality contradicts the definition of $R(t,\omega)$. Consequently, the lexicographic minimum of $K(t,\omega)$, denoted by $x^*(t,\omega)$, equals the lexicographic minimum of $\tilde{K}(t,\omega)$.
To see that the process $t\mapsto x^*(t)$ is progressively measurable, we rely on the following lemma.
Lemma 3. Let $K$ be a non-empty compact set. We denote the lexicographic minimum of $K$ by $x_K^*$. Then the map
$$(\mathcal{K}(\mathbb{T}^2), d_H) \rightarrow (\mathbb{T}^2,|\cdot|), \quad K\mapsto x_K^*$$
is continuous.
Proof. Suppose that $d_H(K_n,K)\rightarrow K$. Given $\epsilon>0$, let $N\in\mathbb{N}$ be sufficiently so that $d_H(K_n,K)\leq \epsilon$ for all $n\geq N$. For each $n\geq N$, there exists $y_{n,K}\in K$ such that
$$|x_{K_n}^*-y_{n,K}| \leq \epsilon,$$
which implies by definition of $d_H$ that
$$x_{K,j}^* \leq x_{K_n,j}^* + \epsilon, \qquad j=1,2.$$
Similarly, there exists $z_{K_n}\in K_n$ such that
$$|x_{K}^* - z_{K_n}| \leq \epsilon,$$
which implies that
$$x_{K_n,j}^* \leq x_{K,j}^* + \epsilon, \qquad j=1,2.$$
Hence, $|x_{K_n}^*-x_{K}^*|\leq 2\epsilon$, and since $\epsilon>0$ was arbitrary, the proof of the lemma is complete. $\Box$
Applying Lemma 3, we conclude that the stochastic process
$t\mapsto x_{\tilde{K}(t)}^*$
is progressively measurable.
