Assume we work under ZFC. Let $(\mathcal X,\rho)$ be a metric space equipped with the Borel $\sigma$-algebra $\mathscr B$ (i.e., the smallest $\sigma$-algebra containing all open sets in $(\mathcal X,\rho)$). Let $$\delta_{\mathcal X} = \sup_{A\subseteq\mathcal X}\{|A|: \text{$A$ is discrete}\}.$$
Assume $\delta_{\mathcal X}\geq \kappa$ where $\kappa$ is a real-valued measurable cardinal. What are necessary / sufficient conditions under which there exists a discrete subset $D\in\mathscr B$ with $|D|=\kappa$?
(As noted below, since $D$ is discrete, one has $2^D\subseteq\mathscr B$ as well.)
not yet an answer
If $A \subseteq X$ then $$ \text{Borel}(A) = \{ A \cap E : E \in \text{Borel}(X)\} . $$
If $A$ is discrete, then $\text{Borel}(A) = 2^A$.
So: if $A \subseteq X$ is discrete, then $2^A \subseteq \text{Borel}(X)$ is equlivalent to $A \in \text{Borel}(X)$.
Then the question becomes: is there a discrete $A \in \text{Borel}(X)$ with $|A| = \kappa$?