Under choice, a space is metrizable if and only if it is paracompact, Hausdorff, and locally metrizable

If $X$ is metrizable, it is Hausdorff and locally metrizable by taking $X$ itself as the open neighbourhood, and it is paracompact by [L1].

Conversely, local metrizability gives an open cover $\mathcal{U}$ by metrizable subspaces. Apply [L3] to obtain a locally finite open cover $\{W_s:s\in S\}$ and assigned $U_s\in \mathcal{U}$ with $\overline{W_s}\subseteq U_s$. Under choice, fix for each $U_s$ a $\sigma$-locally-finite relative open basis ${\bigcup }_{n<\omega }{\mathcal{B}}_{s,n}$ by [L1].

For each $n$, put ${\mathcal{C}}_n=\{B\cap W_s:s\in S,B\in {\mathcal{B}}_{s,n}\}$. Each member is open in $X$, because $U_s$ and $W_s$ are open. The family ${\mathcal{C}}_n$ is locally finite at every $x\in X$: first choose a neighborhood meeting only finitely many $W_s$; for each such $s$, if $x\in U_s$, relative local finiteness of ${\mathcal{B}}_{s,n}$ supplies an ambient neighborhood meeting only finitely many of its members. If $x\mathrm{\notin }{U}_s$, then $x\mathrm{\notin }\overline{W_s}$, so an ambient neighborhood of $x$ misses $W_s$ entirely. Intersect the finitely many chosen neighborhoods.

The union ${\bigcup }_n{\mathcal{C}}_n$ is a basis of $X$: if $x\in O$ with $O$ open, choose $s$ with $x\in W_s$, then choose a relative basis member $B\in {\mathcal{B}}_{s,n}$ with $x\in B\subseteq O\cap U_s$. The open set $B\cap W_s$ contains $x$ and lies in $O$. Thus $X$ has a $\sigma$-locally-finite basis.

Hausdorffness implies $T_1$, and [L2] makes $X$ regular; applying Nagata–Smirnov in [L2] to the basis from step 3.1 yields a metric. Together with step 1.1 this proves the equivalence.