It's known to be consistent with ZF+DC that every subset of $\mathbb{R}$ has the Baire property (BP). (E.g. Shelah's model). If so, then every subset of every complete separable metric space has the BP.

Can we drop the word "separable" here?

Is it consistent with ZF+DC that every subset of every complete metric space has the BP?

In other words, working in ZF+DC, can we prove there exists a complete (non-separable) metric space $X$ and a subset $E \subset X$ without the BP?

I'm not sure which way my intuition goes. On the one hand, non-separable metric spaces are big and maybe they can be weird, even without AC. On the other hand, a counterexample would have the property that the intersection of $E$ with every separable $S \subset X$ would (consistently) have the BP in $S$. That seems unlikely but I don't see how to disprove it.