Update: AJAX to secure servers bug
So, we tried just about everything that can be done within a day (replaced the web service with a proxy ASP page, ran the .send() method in a loop and checked for a valid result, maybe something else ... I don't know. I'm working on 2 hours of sleep), and still we get "The download of the specified resource has failed". I don't believe we've heard back from Microsoft.
If we don't hear back from Microsoft soon, we'll have to yank out AJAX and spend a week rewriting this particular page on the server, which would suck since the rest of the site is AJAX-based. A google search turned up the idea to make sure that there is at least one Language selected in IE. That might enable us to work without having to roll back a 4 month project because of this issue.
Update: We heard back from Microsoft. They've got nothin', as do we. I guess we've got work to do, but that's why we get paid, right?
Update 2: My programmers tell me that Microsoft Support has really pulled out all the stops to help us. Even if they don't find any kind of resolution to this issue, and I suspect that there isn't one, it certainly was not from lack of trying.