Is this restricted promise-3SAT version coNPComplete?
Your problem is coNP-complete when the property is "all variables are zero". Given a SAT formula $\varphi$ on $x_1,\ldots,x_n$, construct a new SAT formula $\psi$ by replacing each clause $C$ of $\varphi$ with $\lnot y \lor C$ (where $y$ is a new variable), and adding clauses $y \lor \lnot x_i$ for $1 \leq i \leq n$. The satisfying assignments of $\psi$ are obtained from those of $\psi$ by setting $y = 1$, and there is an additional satisfying assignment in which all variables are zero. Hence $\varphi$ is satisfiable iff not all satisfying assignments of $\psi$ satisfy the property.