Thanks for all the replies everyone. Keep the suggestions and anecdotes coming. I'm cautiously optimistic that we'll be able to get our visa. If I wasn't concerned about getting denied entry to the country, I would probably go for the airport visa option. But I don't want to misstep on this trip and end up in a place (figuratively or literally) that I don't want to be.

As a side note, I don't want this thread--that I started--to turn into a political discussion. What's happened between the governments has happened, so let's leave that be and keep the discussion on-topic about strategies for U.S. passport holders to get visas and other necessary documentation for visiting Libya.