Well, back to square one it is. Disconnecting the siren didn't help, nor did switching in a known good siren.

Had an idea - is it possible to get the lock/unlock switch (on the console) to work without the ignition switch on?
If so, I could wire in an external switch in parallel to the console switch. Of course, I would only do the "lock" side of the switch and use the door key for unlocking.