I've found the answer - it's NO!

The paper I found addressing the question is the following:

http://projecteuclid.org/euclid.aop/1175287757 ("0-1 Laws for Regular Conditional Probabilities")

A simple counterexample (which I've just slightly adapted from the counterexample in Example 2 of the above paper) is the following:

Let $X=[0,1] \times \{0,1\}$ (with $\Sigma=\mathcal{B}([0,1]) \otimes 2^{\{0,1\}}$), let $\rho$ be a probability measure on $X$ whose projection onto $[0,1]$ is atomless, and let $\mathcal{E} \subset \Sigma$ be the $\rho$-completion of $\mathcal{B}([0,1]) \otimes \{0,1\}$ relative to $\Sigma$. Then given any non-trivial probability measure $m$ on the binary set $\{0,1\}$, the stochastic kernel

 

$\hspace{5mm} \rho_{(x,i)}^\mathcal{E} \ := \ \delta_x \otimes m$

 

is a rcd of $\rho$ with respect to $\mathcal{E}$. Clearly, for any $x \in [0,1]$, $\{(x,0)\} \in \mathcal{E}$; and yet, for all $(x,i) \in X$,

 

$\hspace{5mm} \rho_{(x,i)}^\mathcal{E}(\{(x,0)\}) \ = \ m(0) \, \in \, (0,1).$

Regarding my motivation: Theorem 12 of the above paper claims to be a generalisation of the ergodic decomposition theorem (for measurable maps). However, I haven't yet managed to work out how to derive the ergodic decomposition theorem from Theorem 12 of the above paper.

I've found the answer - it's NO!

The paper I found addressing the question is the following:

http://projecteuclid.org/euclid.aop/1175287757 ("0-1 Laws for Regular Conditional Probabilities")

A simple counterexample (which I've just slightly adapted from the counterexample in Example 2 of the above paper) is the following:

Let $X=[0,1] \times \{0,1\}$ (with $\Sigma=\mathcal{B}([0,1]) \otimes 2^{\{0,1\}}$), let $\rho$ be a probability measure on $X$ whose projection onto $[0,1]$ is atomless, and let $\mathcal{E} \subset \Sigma$ be the $\rho$-completion of $\mathcal{B}([0,1]) \otimes \{0,1\}$ relative to $\Sigma$. Then given any non-trivial probability measure $m$ on the binary set $\{0,1\}$, the stochastic kernel

$\hspace{5mm} \rho_{(x,i)}^\mathcal{E} \ := \ \delta_x \otimes m$

is a rcd of $\rho$ with respect to $\mathcal{E}$. Clearly, for any $x \in [0,1]$, $\{(x,0)\} \in \mathcal{E}$; and yet, for all $(x,i) \in X$,

$\hspace{5mm} \rho_{(x,i)}^\mathcal{E}(\{(x,0)\}) \ = \ m(0) \, \in \, (0,1).$

Regarding my motivation: Theorem 12 of the above paper claims to be a generalisation of the ergodic decomposition theorem (for measurable maps). However, I haven't yet managed to work out how to derive the ergodic decomposition theorem from Theorem 12 of the above paper.