The clicking does sound like a relay tripping in & out, but I've no idea what it may be.
It wouldn't give the "dying after start" symptoms, but I have heard of many immobilisers draining the battery after a time. A low voltage could also lead to relays not holding properly, but once the car fires the alternator should then make up any voltage deficiency if that was the problem. Hmmm (thinks).
ps: You did say CLICKING didn't you (not TICKING! => run away! run away! :green: )