Could you add a comment above the lines defining self._BytesIO, describing why they're being set?  Someone else might see them as unnecessary and rip them out if there's no explanation.

Can a test launch Python in a subprocess, set up the appropriate data structure, let the subprocess Python shutdown, then check the subprocess's stderr for the error?

It's a little convoluted, but perhaps it could be built into a general utility function that could be used to test __del__ methods in other modules, too.

Not sure that it's worth the effort though.