The approach you're taking is much more complicated than the easiest approach to generating the graphs counted by A000109. So if you don't mind, I'll give you an easy-to-implement way to generate those graphs, rather than an easy-to-implement way to check for planarity (which I'm not sure exists,...
^ Very related.
Basically number of planar graph with degree of every vertices >= 4 and deduplicate by spherical isomorphic.
Although, it's quite surprising to me that planarity testing can be done in O(n). Unfortunately graph isomorphism can't be done effectively (with known methods).
One OEIS after another
One OEIS after another