I successfully replaced the alarm siren batteries with an external rechargeable battery (now mounted next to the master cylinder). The printed circuit board in the siren unit was quite badly corroded by the old on-board batteries. This cleaned up ok after considerable scrubbing with alcohol and a dunking in an ultrasonic bath.
The siren now "chirps" once when locking and twice when unlocking. Apparently the alarm LED display on the dashboard should now be out, BUT, in my car, the LED comes back on permanently after the initial alarm system check (all doors, trunk, engine hatch, etc., closed).
Any other reason that is causing the LED to be permanently on?