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In running the test under a local build, the issue is very repeatable, but I believe it's actually due to slow process startup rather than a quick exit.
That is, adding a brief sleep after process creation and just before the Request() call seems to fix the problem. I'm guessing the buildbot is sluggish enough that it just takes a bit longer for the process to start and be ready to be used. 
Whether or not that's purely a machine or test problem, or whether it means that perhaps subprocess.Popen() is returning sooner than it should with a process that isn't ready yet is unclear. (Or even if subprocess can tell)
With the machine fairly idle, it looks like the minimum workable delay is about 200ms - probably something a bit longer would be safer under load, if looking for a quick workaround.