Intermittent ignition problem - immobilizer?

I'm having occasional ignition problems with my 1.6 Aveo5 LS (2012 model year, but not the Sonic). Essentially, some days it will crank fine but never ignite, and this can either mean the car won't start for the next day or it will start 5mins later. There is no pattern to it, except if it starts it will likely start fine for the rest of that day until it sits for longer than 24hrs again. No specific time of day and happens whether it's warm or cold (South African climate means I don't have to worry about sub-zero temps).

I've taken it to a dealership who insist that the problem is with a third-party immobilizer system. I can confirm that it has a third-party alarm but as far as I'm aware these are two different things. I have sent it to two other workshops who have stated it has a factory-fitted immobilizer built into the car, using a transponder system. It has been looked at by all 3 places, and all say the codes are fine and the car is in excellent condition apart from this starting issue (they've checked all the usual culprits of starting issues and it comes up clean).

If it's a built-in system, the dealership should cover it under warranty (it has the last year of warranty left). If it's third-party, I've taken it to two places that couldn't find any sign of a third-party immobilizer. So I either have a lying dealership, incompetent workshops in the area, or a ghost immobilizer...

Can anyone confirm that these keys have a transponder (pic below, with the Autowatch third-party alarm)? They have a red spot on one of the front surfaces that seems to indicate the presence of a chip. Unfortunately I don't have the knowledge to look at this myself and tell the dealer they're talking BS or not. Any other suggestions for what to tell mechanics to look at?

Thanks for any help.

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