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Once the car is fully "On", the 12V battery isn't used - all 12V power comes from the drive battery via the DC-DC converter. If the ICE is on, the car must be fully "On", so that's not your issue. All the 12V battery does is turn on the computers in the car and connect the drive battery into the circuit: if it manages that, the 12V battery should be OK. Thereafter you could almost disconnect it and the car should still run as normal. Obviously yours is misbehaving, as the car will run the ICE when needed to maintain the drive battery, which will in turn should maintain the 12V battery. You need to see a dealer who can run some diagnostics on the car (unless you have a fault-code reader), as the problems you've described mean that the car is definitely not behaving as it should