The Start-Stop on my Parents 12 plate TwinAir Lounge works fine. Occasionally it'll stop and if I've not re-started it after a few minutes it'll beep and say Start & Stop Unavailable, but on the whole it's fine.
My Parents always de-activate Start-Stop but I leave it switched on. However, i'll not let it kick in until a few minutes of gentle driving if I've been on a 70mph run, because I don't think it'll do the turbo any good long term if the engine cuts out straight after a fast run.
Last weekend I took it to London and had it switched on, it cut out at some lights so I thought i'd test the theory about it being linked to the Climate Control - I switched the CC temp up a few degrees and the car re-started automatically, which I thought was really smart and confirms that if the cabin temp is more than a few degrees away from that on the climate control selector, the Start-Stop won't cut in. So definitely no issues here. I'd take issue with an earlier post that Start-Stop should work with the AirCon on. Given that when AirCon is running, most cars will activate the engine fan (at least intermittently) I couldn't expect a battery the size of the 500s supporting the AirCon compressor and the engine fan without getting any charge from the engine, so don't expect it to cut out in this situation.
However, I will say that taking a car back to a dealer 12 times just because the Start-Stop isn't working seems borderline obsessive, and I suffer from fully diagnosed OCD! If the engine was hanging off its mounts I could understand, but it seems like the car essentially works fine, it's just the Start-Stop that doesn't cut in. It saves negligible fuel in most situations & to me seems to be fitted to cars just to satisfy the EU leaf-munching, tree hugging brigade, so I really wouldn't be getting that anxious about it. The only thing i'd do, since the dealer is obviously a bit clueless, is request the battery is replaced under warranty & see if that works. If it doesn't i'd just forget about it. If it does work then happy days, but I would hope that after 12 visits either yourself or the dealer would've thought about a new battery anyway.