You may try sagemanifolds.obspm.fr inside SageMath.

Indeed, my only example in dimension 2 is the triangle with vertices (0,0), (2,0) and (0,2).

Koszul duality of operads involves the inversion of symmetric functions with respect to plethysm. For example, the operads Comm and Lie are Koszul dual. Their associated symmetric functions (recording the action of symmetric groups on components of the operad) are (up to suspension) inverse of each other for the plethysm.

This works for q=4 too, using Peter Taylor code adapted to work over GF(q). The number is 1833840.

Using mutation of quivers with potentials ?