I contacted the registrar and gave her a permission slip so that she could talk to the doctors too. This was partly to circumvent the dean, as he is MIA, and the registrar has some power.

The registrar was more reassuring. She said that I had provided enough information, and that I should be able to take my exam. She CC'd the dean. Now I just need a final thumbs-up from the dean. I hope he's more helpful now, and that the situation is resolved soon.