Let $X$ be a locally compact, locally metrizable space with countable extent, and suppose $X$ has a dense metrizable subspace. Is $X$ separable?
Definition: A space $X$ has countable extent if every uncountable subset of $X$ has a limit point in $X$.
Counterexample. Let $X=\omega_1,$ the set of all countable ordinals, with the order topology.
$\omega_1$ is locally compact and locally metrizable: for each point $\alpha\in\omega_1,$ the segment $[0,\alpha]$ is an open neighborhood of $\alpha$ which is compact and metrizable.
$\omega_1$ has countable extent: if $S$ is an uncountable subset of $\omega_1,$ then $\alpha=\min\{\xi\in\omega_1:\xi\cap S\text{ is infinite}\}$ is a limit point of $S$ in $\omega_1.$
$\omega_1$ has a dense metrizable subspace: the set $Y$ of all isolated points of $\omega_1$ is dense in $\omega_1;$ as a discrete space, $Y$ is metrizable.
$\omega_1$ is not separable: every countable subset of $\omega_1$ has a countable closure.