Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal $\kappa$.

To clarify, $\{0,1\}^\kappa$ is equipped with the usual $\sigma$-algebra for infinite products, which is the smallest $\sigma$-algebra making all component projections $\{0,1\}^\kappa\to\{0,1\}$ measurable. And by "isomorphic to a measurable subset", I mean that there should exist a measurable injection $X\hookrightarrow\{0,1\}^\kappa$ which also takes measurable sets to measurable sets.

For example, such an embedding trivially exists for a standard Borel space, since $\{0,1\}^{\mathbb{N}}$ is itself a standard Borel space.

Clearly, a necessary condition is that $(X,\Sigma)$ needs to be separated in the sense that for all $x,y\in X$, there is $A\in\Sigma$ with $x\in A$ and $y\not\in A$.

Is separatedness sufficient for $(X,\Sigma)$ to embed into some $\{0,1\}^\kappa$?

There always is an obvious measurable map $X\hookrightarrow\{0,1\}^\Sigma$, but I don't even see why it should have measurable image, and I suspect that in general it doesn't. Hence I am at a loss as to how to approach this question. Thanks!