The following modification of the Vietoris topology seems to satisfy the requirements.

Let $\tau$ be the topology on $2^X$ consisting of the sets $\mathcal U\subset 2^X$ such that for every closed set $F\in\mathcal U$ there are open sets $U_1,\dots,U_n\subset X$ intersecting $F$ and a continuous pseudometric $d$ on $X$ such that the set $$\bigcap_{i=1}^n\{E\in 2^X:E\cap U_i\ne\emptyset\}\cap\{E\in 2^X:E\subset \bigcup_{x\in F}B_d(x,1)\}$$ is contained in $\mathcal U$.

Here $B_d(x,1)=\{y\in X:d(x,y)<1\}$ is the open 1-ball cenetered at $x$.