Is there some way to create a vertex attribute that is only attached
to one vertex type in a bipartite graph (in this example the vertices
for which V(g)$type == FALSE)?
Unfortunately there's no way to do that; this is because bipartite graphs in igraph are simply "normal" graphs with a "type" vertex attribute and an additional property that edges never go between vertices of the same type. (But this is not even enforced; you can create such an edge if you really want to). So, for the time being you need to settle with having NA as the "gender" for the company vertices. Try to think about NA as "not applicable", which actually sort-of-makes sense for companies :)
All the best,
Tamas