Is this restricted promise-3SAT version coNPComplete? Given a $3SAT$ problem the decision version of the problem (if it has a solution or not) is $NPComplete$. Now, given that we consider the promise version of the $3SAT$ problem, the promise is every $3SAT$ problem provided always has at least 1 solution.

Query: Does every solution of the above $3SAT$ problem instance have a particular property (one that can be tested in polynomial time. For eg: 'for any solution $A_i$, $N$ numbers immediately smaller than A are co-prime' or something similar; where $N$ is the input length)?

The problem and property can be encoded as a pair consisting of $3SAT$, and another $N$ input circuit of polynomial size ($N^c$ for some constant $c$)

This promise version is definitely in $co-NP$ (as if there is a solution that does not satisfy the given property, that solution is the $co-NP$ certifcate of polynomial length).

Is this problem also $co-NPComplete$ (as it still involves getting the failing solution first)?
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.