Hi Gaius,
I've been guilty of the same style of clumsiness, hence my certainty over how my English teacher would have marked it
I don't really understand your confusion over this, though. It seems pretty clear-cut and unambiguous, to me, even though the second version that you suggested sounds like a more common usage. If someone had written the same sentence in German or Russian, no one would even be quibbling over whether the two names should both go in the genative. The English possessive is essentially the same thing.
Leila's suggestion is a neat alternative, though. That's how you would say it in French, for example.
Alex