Sadly, using the degree sequence as a kind of 'mask' on canonically numbered matrices isn't enough. I should have guessed, it was too simple to be true :(
The smiles for all matrices in the (2, 2, 2, 2, 2, 3, 3) partition are:
and (ignoring the third one, which is disconnected) there are only 3 or 4 different simple graphs in the list.
Perhaps I would have known all this if I knew what automorphisms were, or orbits of elements in sets, or any of that...