Thank you wolf_m! I'm ashamed of myself, I just realized I was being a bad netizen to leave this question hanging out there unanswered...
For anyone interested, here's the solution, provided by most helpful member Evil. Every time before using a context created outside of lwjgl, you must insert a call like this one:
GLContext.useContext(myTrackingObject);
, where myTrackingObject is simply an instance of Object, created by your own code. As Evil explained, that Object serves as a kind of "mark" for the GL context, a place for lwjgl where to put status information that needs to be associated with that context.
So, if you are using several contexts (as I do), you will of course need several of these tracking objects. useContext() must be called right after whatever code you use to actually switch to a new GL context.
A suggestion for the lwjgl makers: wouldn't it be more intuitive to define a dedicated class for this task, or maybe create a factory method ?