You need a decent battery in the fob because there is a proximity sensor surrounding the ignition barrel which senses the fob when you try to start the engine. If it does not sense it then the immobiliser cuts in to stop you starting the engine. Its got nowt to do with the alarm or central locking and pressing any fob buttons.
I know this because mine had exactly the same problem and a new fob battery sorted it. I always carry a spare in me motor as they are dirt cheap and there is no indication that the fob battery is on its way out.