>> (Another note: the C-level errno and the TLS copy should also be
>> synchronized when the C code invokes a Python callback.)
> 
> What I meant is what should occur when a pure Python function is used
> as a callback.  At this point there is already some logic e.g. to
> re-acquire the GIL if necessary.  Maybe it needs to grow logic to
> optionally copy the C-level errno into the TLS variable at the start,
> and at the end copy it back into the C-level errno at the end, for the
> cases where the C code expects the callback to be able to set errno.

I figured that out in the meantime and implemented it in this way.
See the code around line 295 in Modules/_ctypes/callbacks.c.