First, my Greek teacher introduced me to a former student of hers, a Romanian monk who is now on Mount Athos. He is in the Romanian skete near the Great Lavra (Megistis Lavra), and said we could stay one night at the monastery and one night at his skete, if we wanted. (Visitors are generally limited to one night in each location.) See a map here.
When I called around to some of the other monasteries, I was also able to get a hold of someone at Dionysiou, who said we could stay there one night.
Finally, in a long shot, I faxed a request to Simonos Petras and -- against all odds -- we were accepted! (The photo is our 'acceptance' letter.) Since I don't have a fax machine (I can't get the stupid Microsoft Windows Fax to work), I asked Fr. Alexios to send it from the church here in Panorama. He was giving me a hard time about it because he said it was nearly impossible to stay there -- he and Fr. Panoyioti have been trying for a long time and haven't gotten in!
So after talking with a few people about the best ways to get around, the tentative plan is to spend the first night at Dionysiou, the second night at Simonos Petras and the last night at Megistis Lavra before heading back. It should be a great trip, if the weather holds out...