I would be surprised if the alarm were to affect the engine running only when cold; if it were faulty and kicking in the immobilsation circuit at random times, I would have thought it would be when warm as well.
Could the installers have removed/knocked something during the alarm fit?
Do you have a water temperature reading? The cold start map relies on the water temp sensor, so it might be causing a problem?
Have you checked all connections? Coil packs, cam sensors etc?
What about the plugs, are they in good condition? I'm not an expert, but I would have thought that a warm engine would be more forgiving when it comes to ignition than a cold one, so if the plugs are a little dodgy, it might show up more when cold?
Also, what about the idle sensor? Are the pipes on ok?
What is the tickover when cold?
Does it cough or stutter when cold or simply cut out without warning? Also, does this happen at revs or just at tickover?
Just a few thoughts.