Discussing this problem with Alex Ravsky we constructed the following
Example. The Euclidean topology $\tau_0$ on the set $\mathbb Q$ of rational numbers can be enlarged to a regular topology $\tau$ of weight $\omega_1$ such that the countable (and hence cosmic) topological space $(\mathbb Q,\tau)$ is not cometrizable.
The topology $\tau$ is constructed by transfinite induction of length $\omega_1$. At each step $\alpha<\omega_1$ we define a regular second countable topology $\tau_\alpha$ on $\mathbb Q$ such $\bigcup_{\beta<\alpha}\tau_\beta\subset\tau_\alpha\subsetneq \tau_{\alpha+1}$ that the space $(\mathbb Q,\tau_\alpha)$ has no isolated points and for every $\alpha<\omega_1$ the topology $\tau_{\alpha+1}$ contains a neighborhood $U_\alpha$ of zero such that for every $\beta\ge\alpha$ and any neighborhood $V\in\tau_\beta$ of zero the $\tau_\alpha$-closure of $V$ is not contained in $U_\alpha$. Then the topology $\tau=\bigcup_{\alpha\in\omega_1}\tau_\alpha$ is a desired regular topology for which the space $(\mathbb Q,\tau)$ is not cometrizable. $\square$
On the other hand, we have many positive results, see this preprint for more information.
For topological spaces $X,Y$ by $C_k(X,Y)$ we denote the space of continuous functions from $X$ to $Y$, endowed with the compact-open topology.
Theorem 1. For any $\aleph_0$-space $X$ and any cometrizable space $Y$ the function space $C_k(X,Y)$ is cometrizable.
Proof. It is well-known that the $\aleph_0$-spaces are images of metrizable separable spaces under compact-covering maps, which implies that $C_k(X,Y)$ embeds into the function space $C_k(M,Y)$ over some metrizable separable space $M$. So, we can assume that the space $X$ is metrizable and separable. Let $D$ be a countable dense set in $X$. Let $\tau$ be a metrizable topology witnessing that the space $Y$ is cometrizable. It can be shown that the topology on $C_k(X,Y)$ inherited from the Tychonoff power $(Y,\tau)^D$ witnesses that the function space $C_k(X,Y)$ is cometrizable.
Corollary 1. For any $\aleph_0$-space $X$ the function spaces $C_k(X)=C_k(X,Y)$ and $C_k(C_k(X))$ are cometrizable.
A Tychonoff space $X$ is Ascoli if the canonical map $\delta:X\to C_k(C_k(X))$ assigning to each $x\in X$ the Dirac measure $\delta_x:f\mapsto f(x)$ is a topological embedding. It is known that each $k$-space is Ascoli.
The definition of an Ascoli space and Corollary 1 implies
Corollary 2. Each Ascoli $\aleph_0$-space is cometrizable. In particular, each sequential $\aleph_0$-space is cometrizable.
Finally, we have
Theorem 2. Each stratifiable space $X$ is cometrizable.
Proof. Since $X$ is stratifiable, every point $x\in X$ has a decreasing system of open neighborhoods $(W_n(x))_{n\in\omega}$ such that each closed set $F\subset X$ is equal to $\bigcap_{n\in\omega}\overline{W_n[F]}$ where $W_n[F]=\bigcup_{x\in F}W_n(x)$.
It is known that each stratifiable space is a $\sigma$-space. So $X$ has a $\sigma$-discrete network $\mathcal N$, which can be written as the countable union $\mathcal N=\bigcup_{k\in\omega}\mathcal N_k$ of discrete families in $X$.
Since stratifiable spaces are paracompact, each set $N\in\mathcal N_k$ has an open neighborhood $O_N\subset X$ such that the family $(O_N)_{N\in\mathcal N_k}$ is discrete in $X$.
For every $k,n\in\omega$ and $N\in\mathcal N_k$ consider the open neighborhood
$$O_{k,N,n}=O_N\cap W_n[N]$$ of $N$. Since stratifiable spaces are perfectly normal, there exists a continuous function $f_{k,N,n}:X\to [0,1]$ such that $f_{k,N,n}^{-1}(1)=N$ and $f_{k,N,n}^{-1}(0)=X\setminus O_{k,N,n}$.
Let $\ell_1(\mathcal N_k)$ be the Banach space of all functions $h:\mathcal N_k\to\mathbb R$ such that $\|h\|:=\sum_{N\in\mathcal N_k}|h(N)|<+\infty$.
For every $N\in\mathcal N_k$ let $e_N:\mathcal N_k\to\{0,1\}$ be the unique function such that $e_N^{-1}(1)=\{N\}$. So $(e_N)_{N\in\mathcal N_k}$ is the standard unit basis of the Banach space $\ell_1(\mathcal N_k)$.
Consider the continuous function $$f_{k,n}:X\to\ell_1(\mathcal N_k),\;\;f_{k,n}:x\mapsto\sum_{N\in\mathcal N_k}f_{k,N,n}(x)e_N.$$
Next, consider the continuous (injective) function
$$f:X\to\prod_{k\in\omega}\ell_1(\mathcal N_k)^\omega,\;\;f:x\mapsto \big((f_{k,n}(x))_{n\in\omega}\big)_{k\in\omega}.$$
Let $\tau$ be the metrizable topology on $X$ such that the function $f:(X,\tau)\to\prod_{k\in\omega}\ell_1(\mathcal N_k)^\omega$ is a topological embedding.
It follows that for any $k,n\in\omega$ and $N\in\mathcal N_k$ the set $N$ is $\tau$-closed and the set $O_{k,N,n}$ is $\tau$-open.
We claim that the topology $\tau$ witnesses that the space $X$ is cometrizable.
Given any point $x\in X$ and an open neighborhood $O_x\subset X$ of $x$, use the regularity of $X$ to find an open neighborhood $U$ of $x$ such that $\overline{U}\subset O_x$. Consider the closed set $F=X\setminus U$ and observe that $F=\bigcap_{n\in\omega}\overline{W_n[F]}$. Since $x\notin F$, there exists $n\in\omega$ such that $x\notin\overline{W_n[F]}$. Then $V_x:=X\setminus\overline{W_n[F]}$ is an open neighborhood of $x$.
Since $\mathcal N$ is a network, the open set $X\setminus\overline{U}\subset F$ coincides with the union $\bigcup\mathcal N'$ of the subfamily $\mathcal N'=\{N\in\mathcal N:N\subset X\setminus\overline{U}\}$. For every $k\in\omega$ let $\mathcal N_k'=\mathcal N_k\cap\mathcal N'$. Observe that the union $W=\bigcup_{k\in\omega}\bigcup_{N\in\mathcal N'_k}O_{k,N,n}$ is a $\tau$-open set such that $$X\setminus\overline{U}\subset W=\bigcup_{k\in\omega}\bigcup_{N\in\mathcal N'_k}O_{k,N,n}\subset W_n[X\setminus\overline{U}]\subset W_n[F]\subset X\setminus V_x.$$
Then $V_x\subset X\setminus W\subset \overline{U}\subset O_x$ and the $\tau$-closure of the neighborhood $V_x$ is contained in $X\setminus W\subset O_x$. $\square$
