I've put some research in on the topic of graph isomorphism for planar 3-connected graphs, but there are an abundance of algorithms of different restriction, theoretical complexity, and frequency of use and I am having trouble finding one that stands out as:
It's hard to know without understanding the different algorithms myself whether I'm better off with one of the older, more-specialized algorithms for this problem or the newer, more-general ones. Among all possible candidates, which one is/ones are the best fit?
I think Weinberg's algorithm fits the bill.
Easy to understand: compute rotation systems for G and H as byproducts of a planarity testing algorithm. Since G and H are 3-connected, these rotation systems are isomorphic up to interchanging clockwise and counterclockwise if and only if G and H are isomorphic. Choose a dart (= edge with an indicated direction) d in G and try mapping it to all darts e in H (and repeat for the other orientation of H). Since G is connected, all other darts d' can be reached from d by composing the two operations of the rotation system for G. Apply the corresponding operations to e and check whether there is isomorphism.
Maximum clarity: aside from the planarity test, the above is a page of code. Maybe you could reuse someone else's planarity test? There's one in Boost, for example. If not, I still think implementing your own is easier than rewriting nauty.
Good practical performance on small graphs: after planarity testing, Weinberg's algorithm is basically two synchronized depth-first traversals for each dart. The total running time is O(|V|2) with no large constants lurking.