Before I answer your question, let me begin with a brief rant about terminology.  The term "lisse" is bad when applied to vertex algebras:

1. It is French for "smooth", but has nothing to do with smoothness of various geometric objects attached to vertex algebras.  For example, it is not equivalent to smoothness of the associated variety.  People who use it get confused and ask questions like this one.

2. We already have terms that are commonly used: It was originally Zhu's "condition C", but "$C_2$-cofinite" has been standard for about 30 years.

The answer to your question is: a zero dimensional variety has no singularities, so the singularities of $X_V$ can't help you with classification of $C_2$-cofinite vertex algebras.  In most of the interesting examples, $X_V$ is a single point.  More generally, $C_2$-cofinite vertex algebras can't be classified up to isomorphism, because there are too many of them.  See [my answer to this earlier, very similar question][1].

However, there is some work on coarser notions of equivalence, similar to classifying lattices up to genus.  Recall that two lattices are in the same genus if and only if their direct sums with the hyperbolic even unimodular lattice $I\!I_{1,1}$ are isomorphic.  [Moriwaki defines][2] genus equivalence by asserting that two vertex algebras are genus-equivalent if and only if their tensor products with the lattice vertex algebra $V_{I\!I_{1,1}}$ are isomorphic.  However, [Hoehn defines][3] the genus of a regular vertex operator algebra by the pair (modular tensor category, central charge).  These equivalences coincide for lattice vertex algebras, and are hence both generalizations of lattice genera.

  [1]: https://mathoverflow.net/questions/321863/
  [2]: https://arxiv.org/abs/2004.01441
  [3]: https://arxiv.org/abs/1708.05990