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For a measurable skew-product map $\Theta \colon \Omega \times X \to \Omega \times X$ whose state space is a standard Borel space $(X,\mathcal{X})$ and whose base is an invertible mixing dynamical system $(\Omega,\mathcal{F},\mathbb{P},\theta)$, given a $\Theta$-invariant measure $\mu$ on $\Omega \times X$ that projects onto $\mathbb{P}$ with disintegration $(\mu_\omega)_{\omega \in \Omega}$, one can define a "random correlation function" $\rho_{g_1,g_2}(n,\omega)$ for a pair of bounded measurable functions $g_1,g_2 \colon \Omega \times X \to \mathbb{R}$, by \begin{align*} &\rho_{g_1,g_2}(n,\omega) = \\ &\ \left| \int_X g_1\!(\omega,x) \, (g_2 \circ \Theta^n)(\omega,x) \, \mu_\omega(dx) - \int_X g_1(\omega,x) \, \mu_\omega(dx) \int_X g_2(\theta^n\omega,y) \, \mu_{\theta^n\omega}(dy) \right|\text{.} \end{align*} For each $g_1,g_2$, this is well-defined up to $\mathbb{P}$-a.s. equality.

Theorem. A measure-preserving dynamical system $(X,\mathcal{X},\mu_X,T)$ is nicely mixing if and only if for every invertible mixing measure-preserving dynamical system $(\Omega,\mathcal{F},\mathbb{P},\theta)$, taking $\Theta(\omega,x):=(\theta\omega,T(x))$ and $\mu:=\mathbb{P} \otimes \mu_X$, every $A,B \in \mathcal{F} \otimes \mathcal{X}$ has $\rho_{\mathbf{1}_A,\mathbf{1}_B}(n,\,\boldsymbol{\cdot}\,) \overset{\mathbb{P}\text{-a.s.}}{\to} 0$ as $n \to \infty$.

Proof of "only if" direction in Theorem. Suppose that $(X,\mathcal{X},\mu_X,T)$ is nicely mixing, and take any invertible mixing measure-preserving dynamical system $(\Omega,\mathcal{F},\mathbb{P},\theta)$. Let $\mathcal{A}$ be the set of all $A \in \mathcal{F} \otimes \mathcal{X}$ with the property that every $B \in \mathcal{F} \otimes \mathcal{X}$ has $\rho_{\mathbf{1}_A,\mathbf{1}_B}(n,\,\boldsymbol{\cdot}\,) \overset{\mathbb{P}\text{-a.s.}}{\to} 0$ as $n \to \infty$. We will show that

For a measurable skew-product map $\Theta \colon \Omega \times X \to \Omega \times X$ whose state space is a standard Borel space $(X,\mathcal{X})$ and whose base is an invertible measure-preserving dynamical system $(\Omega,\mathcal{F},\mathbb{P},\theta)$, given a $\Theta$-invariant measure $\mu$ on $\Omega \times X$ that projects onto $\mathbb{P}$ with disintegration $(\mu_\omega)_{\omega \in \Omega}$, one can define a "random correlation function" $\rho_{g_1,g_2}(n,\omega)$ for a pair of bounded measurable functions $g_1,g_2 \colon \Omega \times X \to \mathbb{R}$, by \begin{align*} &\rho_{g_1,g_2}(n,\omega) = \\ &\ \left| \int_X g_1\!(\omega,x) \, (g_2 \circ \Theta^n)(\omega,x) \, \mu_\omega(dx) - \int_X g_1(\omega,x) \, \mu_\omega(dx) \int_X g_2(\theta^n\omega,y) \, \mu_{\theta^n\omega}(dy) \right|\text{.} \end{align*} For each $g_1,g_2$, this is well-defined up to $\mathbb{P}$-a.s. equality.

Theorem. A measure-preserving dynamical system $(X,\mathcal{X},\mu_X,T)$ is nicely mixing if and only if for every invertible measure-preserving dynamical system $(\Omega,\mathcal{F},\mathbb{P},\theta)$, taking $\Theta(\omega,x):=(\theta\omega,T(x))$ and $\mu:=\mathbb{P} \otimes \mu_X$, every $A,B \in \mathcal{F} \otimes \mathcal{X}$ has $\rho_{\mathbf{1}_A,\mathbf{1}_B}(n,\,\boldsymbol{\cdot}\,) \overset{\mathbb{P}\text{-a.s.}}{\to} 0$ as $n \to \infty$.

So the theorem holds even if "every invertible measure-preserving dynamical system" is replaced by "every Bernoulli automorphism".

Proof of "only if" direction in Theorem. Suppose that $(X,\mathcal{X},\mu_X,T)$ is nicely mixing, and take any invertible measure-preserving dynamical system $(\Omega,\mathcal{F},\mathbb{P},\theta)$. Let $\mathcal{A}$ be the set of all $A \in \mathcal{F} \otimes \mathcal{X}$ with the property that every $B \in \mathcal{F} \otimes \mathcal{X}$ has $\rho_{\mathbf{1}_A,\mathbf{1}_B}(n,\,\boldsymbol{\cdot}\,) \overset{\mathbb{P}\text{-a.s.}}{\to} 0$ as $n \to \infty$. We will show that

For example, Ronnie Pavlov has shown in the comments that a two-sided Bernoulli shift has re-orderable correlations (with $\mathcal{Y}$ the set of cylinder sets and with $\delta(\varepsilon)=\varepsilon$), and so a two-sided Bernoulli shift fulfils the property described in my question.