Such a hypothetical Baire theorem is not true: for every cardinal $\kappa$ of uncountable cofinality and any cofinal subset $C\subset \kappa$ of cardinality $|C|=\mathrm{cf}(\kappa)$, the Hilbert space $\ell_2(\kappa)$ of density $\kappa$ is a complete metric space that can be written as the union $\bigcup_{\alpha\in C}\ell_2(\alpha)$ of $\mathrm{cf}(\kappa)$ many closed nowhere dense subsets $\ell_2(\alpha)=\{x\in\ell_2(\kappa):x^{-1}(\mathbb R\setminus\{0\})\subset [0,\alpha)\}$. Then the intersection $\bigcap_{\alpha\in C}U_\alpha$ of the dense open sets $U_\alpha=\ell_2(\kappa)\setminus\ell_2(\alpha)$ is empty.

The same concerns the closed unit ball $B$ of $\ell_2(\kappa)$. In the weak topology, $B$ is (uniform Eberlein) compact, which can be written as the union of $\mathrm{cf}(\kappa)$ many closed nowhere dense subsets. So, the Baire Theorem does not extend even to the intersection of $\aleph_1$ many dense open sets in nice (namely, uniform Eberlein) compact spaces.

On the other hand, there are some extensions of the Baire Theorem for topological spaces of countable cellularity, but a metrizable space has countable cellularity if and only if it is separable, so such extensions do not concern metrizable spaces. Namely, the Martin's Axiom is equivalent to the statement:

For any compact Hausdorff space $K$ of countable cellularity, the intersection of less than continuum many dense open sets in $K$ is not empty.

As a first reading of this topic, you can look at section 1 called "Topology of MA" of the survey paper "Versions of Martin's Axiom" of William Weiss in "Handbook of Set-Theoretic Topology" published by Elsevier in far 1984.

Also there exists a well-known cardinal characteristic $\mathrm{cov}(\mathcal M)$ of the continuum, equal to the smallest number of open dense subsets in the real line whose intersection is empty. The value of $\mathrm{cov}(\mathcal M)$ is between $\omega_1$ and $\mathfrak c$, but its exact position in the segment
$[\omega_1,\mathfrak c]$ depends on additional axioms of Set Theory. The cardinal $\mathrm{cov}(\mathcal M)$ is one of 10 cardinal characteristics of the continuum, composing the famous Cichon's Diagram.