Pick $\dim V=2$, $G$ trivial, and $\kappa$ a symplectic form. Then $A$ is the first Weyl algebra, which does not contain any commutative algebra isomorphic to $S(V)$. The answer to your *further problem* is thus no.

The algebras you mention in your *related problem* are indeed worthy of study! They are deformations of $S(V)\\#\mathbb CG$, and some of them are even PBW deformations. You can write down the explicit condition for this to happen by unrolling the so called Jacobi condition; you'll find this discussed, for example, in Roland Berger and Victor Ginzburg's paper on non homogeneous PBW deformations of $N$-Koszul algebras (you case is quadratic, so simpler, but they discuss it it)

As for your *pedantic point*, I don't know. I guess you want someting like the symmetrization map that exists for enveloping algebras?