Improvements of the Baire Category Theorem under (not CH)?

A complete space without isolated points has at least continuum cardinality. At least if you agree to use (some form of) Axiom of Choice.

Choose two disjoint closed balls $B_1$ and $B_2$. Inside $B_1$, choose disjoint closed balls $B_{11}$ and $B_{12}$. Inside $B_2$, choose disjoint closed balls $B_{21}$ and $B_{22}$. And so on. At $n$th step, you have $2^n$ disjoint balls indexed by binary words of length $n$, and you choose two disjoint balls of level $n+1$ inside each ball of level $n$. This is possible because the balls are not single points. Make sure that radii go to zero. Now you have continuum of sequences of nested balls each having a common point.

The number of meager sets needed to cover the real line is a "cardinal invariant of the continuum". It is one of the invariants in Cichoń's diagram. In particular, it is Cov(K) in Cichoń's diagram on Wikipedia.

Looking at nowhere-dense sets instead of meager sets would not change this invariant, because of basic cardinal arithmetic.

I am not certain, off the top of my head, if the invariants are the same for every uncountable complete separable metric space.