In the updated version of my previous post, I mentioned nine potentially-minimal forbidden minors for the apex graphs: the seven graphs of the Petersen family, a cube with two doubled vertices, and the double pyramid over a triangular prism.

Unless I've made a mistake in my calculations (entirely likely), the following nine graphs are also minimal forbidden minors for the apex graphs:

Nine more forbidden minors for the apex graphs

In addition, the three components in the three graphs on the right can be mixed and matched, leading to another ten combinations not shown. With the number of obstacles growing to at least 28, my hope for a clean characterization is diminishing.

ETA: Still another one: start with a cube, find a four-vertex independent set, and make three copies of each of its vertices. The resulting 16-vertex graph has four \( K_{3,3} \) subgraphs, one for each tripled vertex.