This one's a real head-spinner. For some reason I've yet to be able to determine, the queue would crash whenever I pushed jobs that rebuilt SSIs for a template. (I have several in the template for this site right now.) But if I saved and rebuilt the SSI directly from the template editor, no crash!
I still don't know why this is happening, but I've found that by setting the build priority for SSIs to something higher than other job types, the crash goes away. What's more, the crash was not one I've ever seen with this project before -- a SQLite error, "unable to close due to unfinalised statements." I spent an hour trying to figure out where it might be leaking in before discovering the fix.
(SSIs are listed in the queue as "include files". Note the priority of "10", where other jobs are "9". That fixed the problem.)
Here's the thing: while that may ameliorate the bug, it hasn't told me why it exists. To my mind, the latter is far more important than the former. If I don't know why this happened, or why changing the build priority fixes it, I'm likely to only repeat the same mistake further on down the line in another form.
For the time being, though, at least I have it fixed -- and I'm now able to have SSIs properly rebuilt whenever a page is saved.