variant of Critical SAT
It is clear that your language is in DP. In order to show that it is DP-hard, we will give a reduction from SAT-UNSAT to your language, which we can call CRIT-UNSAT. Given a pair of CNFs $(f,g)$, let $x,y$ be fresh variables, and let
$$
h = (f \lor \lnot x) \land (g \lor x) \land (g \lor y) \land \lnot x \land (x \lor \lnot y).
$$
Here $f \lor \lnot x$ means add $\lnot x$ to all clauses of $f$.

Suppose first that $f$ is satisfiable and $g$ is not satisfiable. Since $g$ is not satisfiable, $h$ is not satisfiable. Since $f$ is satisfiable, $h \setminus \lnot x$ is satisfiable. Thus $h$ is in CRIT-SAT.

Conversely, suppose that $h$ is in CRIT-SAT. Since $h$ is unsatisfiable, $g$ is unsatisfiable. For some clause $c$, $h \setminus c$ is satisfiable. If $c \in f \lor \lnot x$ then clearly $h \setminus c$ is still unsatisfiable. Similarly, if $c \in g \lor x$ then $h \setminus c$ is still unsatisfiable, due to $g \lor y$. If $c \in g \lor y$ or $c = x \lor \lnot y$ then $h \setminus c$ is still unsatisfiable, due to $g \lor x$.
Thus $c = \lnot x$, which means that $h|_{x=1}$ is satisfiable, that is, $f$ is satisfiable.