What happens when the battery dies?

BTW, does the design allow for the battery to be replaced without turning the system off (assuming said replacement is feasible (i.e. no perimeter protection in place) and done with all due precautions (e.g. no metallic tools and ESD grounding)?

Yes, If the host power remains ON, (and your perimeter detect is disabled so you can get access to the battery), then you can replace the battery without loosing RTC or perimeter detect functions.