I put the new battery in.
The car exhibited the wonky behaviour I've described before including magically unlocking doors. After I turned the anti-theft system on & off a few times and started / restarted the car and left it idling for 5 minutes it settled down and acted normally. It was just fine on the road and was still okay when I got back home.
Took readings a few times & about. Battery charge is normal, alternator reading is normal (measured across the battery) while the car is revved up to 1500, and also running at idle.
I'm still thinking it has something to do with the anti-theft system OR maybe a relay under the dash - power door lock relay maybe? I heard something click right before the doors unlocked themselves.
While out we didn't use the power door locks as a precaution.
Will do some reading about power door lock relays etc. They are way up high under the dash and a real pain to get at.
Will likely do an amp draw test later today too to see if there's a parasitic drain.