The post would still go through, and everything works fine - except that I end up with an error in the logs, and the status of that post is an error message rather than "Yay! You posted a thing!".
So, on the grounds that my code is perfect, I assumed that Wordpress.com was somehow doing something that timed me from there end. Clearly, in retrospect, this was a very stupid thing to think.
Because the _actual_ answer is that the default timeout for a URL fetch from Google App Engine is 5 seconds. And the WordPress site is taking about eight seconds to make the post and then tell me it has.
And there's no way for me to change the default. And the library I'm using lets you change the timeout for some transports, but not the one that AppEngine supports - even though App Engine supports setting the timeout.
And it turns out that this project, which is the one that all Google searches take you to, if you search for "XML-RPC Java", is no longer supported, and hasn't been updated since 2010.
So my options are:
1) Download the source code for an unsupported library and update it myself.
2) Switch libraries.
Both of which sound awful, but the latter sounds slightly less awful.
I may spend half an hour being repeatedly killed by Manus instead.
Original post on Dreamwidth - there are comments there.